Method and apparatus for accurate, micro-contact printing

ABSTRACT

Disclosed is a printing apparatus, having a print surface lying in a print plane defined by an imaginary x-axis and y-axis, the print surface having an outward normal pointing in the positive direction along an imaginary z-axis, such that the x-axis, y-axis, and z-axis are substantially orthogonal to one another, a lower stamp clamp disposed adjacent to the negative-x edge of the print surface, an upper stamp clamp, moveable in two dimensions in a trajectory plane defined by the x-axis and z-axis, a stamp comprising a flexible material, the stamp having a first end attached to the lower stamp clamp and a second end attached to the upper stamp clamp, such that a cross section of the stamp parallel to the trajectory plane forms an arc extending from an origin point Q on the lower stamp clamp having (x,z) coordinates (0,0) to point E on the upper stamp clamp, this arc being described by the mathematical function θ(s), where s is the curvilinear distance along the arc measured from point Q, and θ is the angle between the print plane and an imaginary line, the imaginary line being tangent to the cross section of the stamp at s, and wherein, during a print operation, the upper stamp clamp is moved in a trajectory comprising a plurality of xz positions of the upper clamp stamp that blend into a substantially continuous motion, the trajectory being effective in laying the stamp down smoothly and flat upon the print surface in a manner such that a moving contact front between the stamp and the print surface is created, the contact front being disposed substantially along a line characterized by a contact-front coordinate s 0  x 0  that increases as the trajectory progresses, the trajectory also being effective in causing the curvature dθ/ds of the stamp at or near the contact front to be substantially constant throughout the motion. The preferred embodiment also has either or both of two additional systems that move along the x axis in coordination with the motion of the upper stamp clamp: a print-force-application system effective in pressing the stamp against the print surface, and a stamp control system for helping to control the curvature of the stamp near the contact front.

FIELD OF THE INVENTION

[0001] The invention relates to printing, particularly to micro-contactprinting, also called “soft lithography”, in which a flexible stamptransfers an “inked” pattern to a receiving surface by mechanicalcontact, the pattern often having very small features normallyassociated with optical lithography and other expensive methods. Moregenerally, the invention relates to a precise and controlled way ofbringing two surfaces into contact, and subsequently separating them.

BACKGROUND OF THE INVENTION

[0002] A number of printing techniques collectively known as “softlithography” have been recently developed, spurred by the 1993 discoveryof micro-contact printing, as described in A. Kumar and G. M.Whitesides, FEATURES ON GOLD HAVING MICROMETER TO CENTIMETER DIMENSIONSCAN BE FORMED THROUGH A COMBINATION OF STAMPING WITH AN ELASTOMERICSTAMP AND AN ALKANETHIOL INK FOLLOWED BY CHEMICAL ETCHING, Appl. Phys.Lett., 63, 2002 (1993), the disclosure of which is incorporated byreference herein in their entirety. Typically, in such a printingtechnique, a flexible, polymeric stamp, embossed with a pattern andcoated with a chemical “ink”, is brought into contact with a receivingsurface and then separated from it, thereby transferring the image tothe receiving surface in the form of a molecular monolayer of the ink. Afull review of the techniques of soft lithography has recently beengiven in Y. Xia and G. M. Whitesides, SOFT LITHOGRAPHY, Angew. Chem.Int. Ed., 37, 550 (1998) and in B. Michel, et al., PRINTING MEETSLITHOGRAPHY: SOFT APPROACHES TO HIGH RESOLUTION PATTERNING, to bepublished in IBM Journal of Research and Development (special issue onlithography), the disclosures of both of which are incorporated byreference herein in their entirety.

[0003] Soft lithography promises to deliver printing that is less costlythan that available with conventional techniques, such as opticallithography, used routinely in semiconductor processing. Softlithography's lower cost is possible because the per-print process issimpler than conventional techniques—there are fewer steps and fewercostly machines. Moreover, soft lithography can print large areasquickly, whereas traditional, optical techniques can print only smallareas at a time, and must build up large areas by “stitching” (step andrepeat), a slow process requiring an extremely precise and expensivemachine known as a lithographic stepper.

[0004] To enable soft lithography, a printing method and apparatus arerequired to bring the stamp and the receiver into intimate contact, in acontrolled and repeatable manner, such that the pattern on the stamp istransferred to the receiver with the greatest possible fidelity (i.e.,with minimal distortion). To insure intimate contact everywhere, theprinting method must prevent the trapping of gaseous bubbles (e.g., airbubbles) between the opposing surfaces of the stamp and the receiver. Toinsure repeatability, the printing apparatus must be automated. Toachieve high fidelity, two requirements must be met. Firstly, the stampitself should resist distortions in its own plane; such resistance isprovided, for example, by the two-layer “hybrid stamps” described by B.Michel et. al., supra. Secondly, the printing apparatus must provide,when the stamp and the receiver come into contact, uniform contactpressure and uniform geometric conditions over the entire printed area,lest the stamp be non-uniformly strained and therefore the printedpattern distorted.

[0005] Several prior-art methods and machines attempt to provide theprinting requirements needed for soft lithography. However, theseprior-art methods are deficient in several respects. One such method isdescribed in U.S. Pat. No. 5,669,303 entitled APPARATUS AND METHOD FORSTAMPING A SURFACE, issued Sep. 23, 1997. This apparatus brings acircular stamp, held at its edges, into gradual contact with a receiver.The stamp is treated as a membrane under variable pressure: the convex(lower-pressure) side of the curving stamp being gradually flattenedagainst the receiver while the periphery of the stamp is held fixed.Although the gradual contact successfully eliminates the trapping of airbubbles, this method and apparatus clearly produces non-uniform strainin the stamp as the varying pressure stretches the membrane, therebydistorting the pattern. Acknowledging this distortion, various schemeswere proposed to compensate it, but the manufacturing practicality ofthese schemes is doubtful, and it would clearly be preferable if themethod did not engender the non-uniform strain in the first place.

[0006] Another prior-art apparatus and method are described by B. Michelet al., supra, as the “rocker cylinder printing tool”. In this method,the stamp is wrapped on a partial drum of radius R, and then “rocked”upon the receiving surface in a manner somewhat analogous to the motionof a rocking chair upon a floor. In other words, the method is like aprinting press in which the receiver remains stationary while the axisof the rotating drum translates over it. The problems with this methodare three-fold. Firstly, the embossed pattern on the stamp is stretchedin the print direction due to the drum's curvature, introducingsystematic distortion. Secondly, over the print cycle, the peak contactpressure between the stamp and the receiver is spatially non-uniformbecause it depends critically on the drum-to-receiver gap, which variesas the mechanism moves on account of unavoidable mechanical tolerancessuch as bearing runout and machining inaccuracies on the drum's surface.Attempting to minimize variations in peak contact pressure byintroducing a compliant layer (known as a “soft pad”) behind the stampsimply trades peak-pressure non-uniformity for geometric non-uniformity;that is, as the soft pad compresses to accommodate gap changes, thelocal curvature of the stamp near the line of contact varies, and thusthe tangential strain of the embossed pattern varies—this complexvariation being superimposed on the systematic strain due to the drum'scurvature. Thirdly, because the drum is both translating and rotating,the accuracy of printing depends critically on precisely matching thedrum's translational speed ν with its rotational speed ω; ideally, toroll without slipping and without straining the compliant stamp byfrictional forces, the drum's velocity ν should be exactly equal to ωR.However, this ideal matching is nearly impossible to accomplish to thetolerance (˜1 ppm) required for high-accuracy, large-areaapplications—exactly the applications where soft lithography seeks toreplace optical lithogtaphy. Thus the rocker-style printer is ill-suitedto the task of soft lithography. In fact, a controlled experiment wasperformed in which feature-placement errors on two prints from the samestamp were measured—one print made with a well-engineered rockerprinter, the other with an alternative scheme (such as the currentinvention), where the three problems discussed above are absent. Theresults demonstrate roughly a factor-of-three advantage infeature-placement accuracy for the latter method.

[0007] All three shortcomings of the rocker printer, of course, areshared by the “printing press” style of machine. In particular, theprinting press shares the third shortcoming mentioned above (printaccuracy dependent on precise matching of ν to ωR): although theprinting-press's drum rotates without translating, the receiver insteadtranslates beneath it, so speed matching is still an issue. Although theprinting press is, of course, suitable for images to be observed by thehuman eye, where feature-placement accuracy need not be better thanabout 10 to 20 μm, it appears to be unsuitable for the applications ofsoft lithography (e.g., printing patterns for electronic circuitry),where feature-placement accuracy on the order of 1 μm or better isrequired.

[0008] Accordingly, there is a need for an improved method and apparatusfor transferring patterns from a stamp to a receiver with greatfidelity, the method and apparatus being scalable to large-sizereceivers and amenable of various types of stamps.

SUMMARY OF THE INVENTION

[0009] Disclosed is a printing apparatus, comprising a print surfacelying in a print plane defined by an imaginary x-axis and y-axis, theprint surface having an outward normal pointing in the positivedirection along an imaginary z-axis, such that the x-axis, y-axis, andz-axis are substantially orthogonal to one another, a lower stamp clampdisposed adjacent to the negative-x edge of the print surface, an upperstamp clamp, moveable in two dimensions in a trajectory plane defined bythe x-axis and z-axis, a stamp comprising a flexible material, the stamphaving a first end attached to the lower stamp clamp and a second endattached to the upper stamp clamp, such that a cross section of thestamp parallel to the trajectory plane forms an arc extending from anorigin point Q on the lower stamp clamp having (x,z) coordinates (0,0)to point E on the upper stamp clamp, this arc being described by themathematical function θ(s), where s is the curvilinear distance alongthe arc measured from point Q, and θ is the angle between the printplane and an imaginary line, the imaginary line being tangent to thecross section of the stamp at s, and wherein, during a print operation,the upper stamp clamp is moved in a trajectory comprising a plurality ofxz positions of the upper clamp stamp that blend into a substantiallycontinuous motion, the trajectory being effective in laying the stampdown smoothly and flat upon the print surface in a manner such that amoving contact front between the stamp and the print surface is created,the contact front being disposed substantially along a linecharacterized by a contact-front coordinate s₀ x₀ that increases as thetrajectory progresses, the trajectory also being effective in causingthe curvature $\frac{\theta}{s}$

[0010] of the stamp at or near the contact front to be substantiallyconstant throughout the motion.

[0011] Another aspect of the printer comprises a print-force-applicationsystem effective in pressing the stamp against the print surface, anddefining an approximate contact front disposed substantially along aline_(B) parallel to the y-axis in the xy plane, the line_(B)intersecting the trajectory plane at (x,z)=(x_(B),0), theapproximate-contact-front x-coordinate x_(B) increasing as thetrajectory progresses and being substantially equal, at any stage of thetrajectory, to the arc-length coordinate s_(B) of point B, inasmuch asthe arc of the stamp is assumed to be substantially flat over thesegment from point Q to point B.

[0012] Another aspect of the printer further comprises a stamp-controlsystem movable along the x-axis, wherein, throughout the trajectory,each xz position of the upper stamp clamp is a function of thedisplacement x_(C) of the stamp-control system along the x-axis; thetrajectory being effective in laying the stamp down upon the printsurface such that the stamp is in continuous contact with a contactsurface of the stamp-control system throughout the trajectory, thelocation of the contact surface being characterized by an arc-lengthcoordinate s_(C) that increases as the trajectory progresses.

[0013] In another aspect of the printer, the stamp-control system isdisposed along a line_(C) parallel to the y-axis, line_(C) intersectingthe trajectory plane at point C having coordinates x_(C) and z_(C),where z_(C) is a fixed, positive z-coordinate during any one printingoperation, whereas x_(C) increases as the trajectory progresses, incoordination with the contact-front coordinate x₀.

[0014] In another aspect of the printer, the contact surface of thestamp-control system is a plane delimited in the x direction by twolines _(C) and _(D) separated by a fixed distance W_(CD), these linesbeing parallel to the y-axis and intersecting the trajectory plane atpoints C and D respectively, these points having coordinates(x_(C),Z_(C)) and (x_(D),z_(D)) respectively, such that the contactsurface is defined by the three parameters (x_(C),z_(C),θ_(CD)), where$\theta_{CD} \equiv {\tan^{- 1}( \frac{z_{D} - z_{C}}{x_{D} - x_{C}} )}$

[0015] is the angle between the contact surface and the print plane, andsuch that the stamp angle θ(s) between arc-length coordinates s=s_(C)and s=s_(D) is substantially equal to θ_(CD); that is,

θ(s)≈θ_(CD) for s _(C) ≦s≦s _(D).

[0016] In another aspect of the printer, the upper stamp clamp ispivoted about a pivot line_(P) parallel to the y axis and intersectingthe xz plane at point P having coordinates x_(P) and z_(P); the stampattaching to the upper stamp clamp along an upper-clamp line E parallelto the y axis and intersecting the xz plane at point E havingcoordinates x_(E) and z_(E); the upper-clamp line_(E) being disposed onthe upper stamp clamp at a radius R_(s) from the pivot line_(P), suchthat the total arc length s_(E) from the lower stamp clamp to theline_(E) is s_(E) L, where L is the known, free length of the stamp; andwherein the stamp attaches to the upper-clamp line_(E) at an angleθ_(E)≡θ(L).

[0017] In another aspect of the printer, the trajectory comprises aplurality of configurations, each configuration described by thecoordinate s₀ x₀ of the contact front and by corresponding coordinatesx_(P), z_(P) of the pivot line given by the equations

x _(P) =x _(E) +R _(s) cos θ_(E)

z _(P) =z _(E) +R _(s) sin θ_(E),

[0018] where x_(E)=∫₀ ^(L) cos θ(s)ds and z_(E)=∫₀ ^(L) sin θ(s)ds, andwhere the mathematical function θ(s) describing the shape of the arc fora given configuration is assumed to be

θ(s)=0 for 0≦s≦s ₀,

[0019] whereas for s>s₀, θ(s) is determined by solution of thedifferential equations ${\frac{u}{s} = {F(u)}},$

[0020] the lower-end boundary conditions ${{u_{0} \equiv \begin{Bmatrix}u_{10} \\u_{20}\end{Bmatrix} \equiv \begin{Bmatrix}\theta_{0} \\ \frac{\theta}{s} |_{0}\end{Bmatrix}} = \begin{Bmatrix}0 \\\kappa_{0}\end{Bmatrix}},$

[0021] and the upper-end boundary condition${ {{T(\beta)} \equiv {E\quad I\frac{\theta}{s}}} \middle| {}_{E}{{{+ F_{X0}}R_{S}\sin \quad \theta_{E}} - {{w( {s - s_{0}} )}R_{s}\cos \quad \theta_{E}}}  = 0},{wherein}$${u \equiv \begin{Bmatrix}u_{1} \\u_{2}\end{Bmatrix} \equiv \begin{Bmatrix}\theta \\\frac{\theta}{s}\end{Bmatrix}},{{F(u)} \equiv \begin{Bmatrix}u_{2} \\{{\frac{F_{X0}}{E\quad I}\sin \quad u_{1}} - {\frac{w( {s - s_{0}} )}{E\quad I}\cos \quad u_{1}}}\end{Bmatrix}},$

[0022] κ₀ is a specified curvature at point O, the parameter β≡F_(X0),unknown a priori, is the internal x-directed force acting on the stamp'scross section at s=s₀ per unit depth of the stamp in the y direction, Eis Young's modulus of the stamp, I is the area moment of inertia of thestamp's cross section per unit depth in the y-direction, and w is theweight per unit area of the stamp; and

[0023] wherein for each configuration the solution for x_(P) and z_(P)is derived by means of the “shooting method”, whereby an initial valueβ⁽⁰⁾ of β is guessed, the differential equations are solved to yieldT(β⁽⁰⁾) and$\lbrack \frac{\partial T}{\partial\beta} \rbrack_{\beta = \beta^{(0)}}^{- 1},$

[0024]  Newton iteration$\beta^{({n + 1})} = {\beta^{(n)} - {\lbrack \frac{\partial T}{\partial\beta} \rbrack_{\beta = \beta^{(0)}}^{- 1}{T( \beta^{(n)} )}}}$

[0025]  is applied to obtain a refined value β⁽¹⁾ of the unknownparameter β, whereupon the differential equations are solved again; thisiteration procedure being applied repeatedly until the correct auxiliaryboundary condition T(β)=0 is achieved to within some tolerance.

[0026] In another aspect of the printer the upper stamp clamp is pivotedabout a pivot line_(P) parallel to the y axis and intersecting the xzplane at point P having coordinates x_(P) and z_(P); the stamp attachingto the upper stamp clamp along an upper-clamp line_(E) parallel to theyaxis and intersecting the xz plane at point E having coordinates x_(E)and z_(E); the upper-clamp line_(E) being disposed on the upper stampclamp at a radius R_(S) from the pivot line_(P), such that the total arclength s_(E) from the lower stamp clamp to the line_(E) is s_(E) L,where L is the known, free length of the stamp; and wherein the stampattaches to the upper-clamp line_(E) at an angle θ_(E)≡θ(L).

[0027] In another aspect of the printer, the trajectory comprises aplurality of configurations, each configuration described by thecoordinate s_(B) x_(B) of the approximate contact front and bycorresponding coordinates x_(P), z_(P) of the pivot line given by theequations

x _(P) =x _(E) +R _(s) cos θ_(E)

z _(P) =z _(E) +R _(s) sin θ_(E),

[0028] where x_(E)=∫₀ ^(L) cos θ(s)ds and z_(E)=∫₀ ^(L) sin θ(s)ds, andwhere the mathematical function θ(s) describing the shape of the arc fora given configuration is assumed to be

θ(s)=0 for 0≦s≦s _(B),

[0029] whereas for s>s_(B), θ(s) is determined by solution of thedifferential equations ${\frac{u}{s} = {F(u)}},$

[0030] the lower-end boundary conditions ${{u_{B} \equiv \begin{Bmatrix}u_{1B} \\u_{2B}\end{Bmatrix} \equiv \begin{Bmatrix}\theta_{B} \\ \frac{\theta}{s} |_{B}\end{Bmatrix}} = \begin{Bmatrix}0 \\\kappa_{B}\end{Bmatrix}},$

[0031] and the upper-end boundary condition${ {{T(\beta)} \equiv {E\quad I\frac{\theta}{s}}} \middle| {}_{E}{{{+ F_{XB}}R_{S}\sin \quad \theta_{E}} - {{w( {s - s_{B}} )}R_{s}\cos \quad \theta_{E}}}  = 0},{wherein}$${u \equiv \begin{Bmatrix}u_{1} \\u_{2}\end{Bmatrix} \equiv \begin{Bmatrix}\theta \\\frac{\theta}{s}\end{Bmatrix}},{{F(u)} \equiv \begin{Bmatrix}u_{2} \\{{\frac{F_{XB}}{E\quad I}\sin \quad u_{1}} - {\frac{w( {s - s_{B}} )}{E\quad I}\cos \quad u_{1}}}\end{Bmatrix}},$

[0032] κ_(B) is a specified curvature at point B, the parameterβ≡F_(XB), unknown a priori, is the internal x-directed force acting onthe stamp's cross section at s=s_(B) per unit depth of the stamp in they direction, E is Young's modulus of the stamp, I is the area moment ofinertia of the stamp's cross section per unit depth in the y-direction,and w is the weight per unit area of the stamp; and

[0033] wherein for each configuration the solution for x_(P) and z_(P)is derived by means of the “shooting method”, whereby an initial valueβ⁽⁰⁾ of β is guessed, the differential equations are solved to yieldT(β⁽⁰⁾) and$\lbrack \frac{\partial T}{\partial\beta} \rbrack_{\beta = \beta^{(0)}}^{- 1},$

[0034]  Newton iteration$\beta^{({n + 1})} = {\beta^{(n)} - {\lbrack \frac{\partial T}{\partial\beta} \rbrack_{\beta = \beta^{(n)}}^{- 1}{T( \beta^{(n)} )}}}$

[0035] is applied to obtain a refined value β⁽¹⁾ of the unknownparameter β, whereupon the differential equations are solved again; thisiteration procedure being applied repeatedly until the correct auxiliaryboundary condition T(β)=0 is achieved to within some tolerance.

[0036] In another aspect of the printer, the upper stamp clamp ispivoted about a pivot line_(P) parallel to the y axis and intersectingthe xz plane at point P having coordinates x_(P) and z_(P); the stampattaching to the upper stamp clamp along an upper-clamp line_(E)parallel to the y axis and intersecting the xz plane at point E havingcoordinates x_(E) and z_(E); the upper-clamp line_(E) being disposed onthe upper stamp clamp at a radius R_(S) from the pivot line_(P), suchthat the total arc length s_(E) from the lower stamp clamp to theline_(E) is s_(E) L, where L is the known, free length of the stamp; andwherein the stamp attaches to the upper-clamp line_(E) at an angleθ_(E)θ(L).

[0037] In another aspect of the printer, the trajectory comprises aplurality of configurations, each configuration described by thecoordinate s₀ x₀ of the contact front and by corresponding coordinatesx_(P), z_(P) of the pivot line given by the equations:

x _(P) =x _(E) +R _(s) cos θ_(E)

z _(P) =z _(E) +R _(s) sin θ_(E),

[0038] where x_(E)=∫₀ ^(L) cos θ(s)ds and z_(E)=∫₀ ^(L) sin θ(s)ds, andwhere the mathematical function θ(s) describing the shape of the arc fora given configuration is assumed to be

θ(s)=0 for 0≦s≦s ₀,

[0039] whereas for s>s₀, θ(s) is determined by solution of thedifferential equations ${\frac{u}{s} = {F(u)}},$

[0040] the lower-end boundary conditions ${{u_{0} \equiv \begin{Bmatrix}u_{10} \\u_{20} \\u_{30} \\u_{40}\end{Bmatrix} \equiv \begin{Bmatrix}\theta_{0} \\ \frac{\theta}{s} |_{0} \\F_{X_{0}} \\F_{Z_{0}}\end{Bmatrix}} = \begin{Bmatrix}0 \\0 \\F_{X_{0}} \\F_{Z_{0}}\end{Bmatrix}},$

[0041] and the auxiliary boundary conditions

T(β)=0,

[0042] wherein ${u \equiv \begin{Bmatrix}u_{1} \\u_{2} \\u_{3} \\u_{4}\end{Bmatrix} \equiv \begin{Bmatrix}\theta_{0} \\\frac{\theta}{s} \\{F_{x}(s)} \\{F_{z}(s)}\end{Bmatrix}},{{F(u)} \equiv \begin{Bmatrix}u_{2} \\{{\frac{u_{3}}{E\quad I}\sin \quad u_{1}} - {\frac{u_{4}}{E\quad I}\cos \quad u_{1}}} \\{{{- {p(s)}}\sin \quad u_{1}} - {{f(s)}\quad \cos \quad u_{1}}} \\{w + {{p(s)}\quad \cos \quad u_{1}} - {{f(s)}\quad \sin \quad u_{1}}}\end{Bmatrix}},$

${{T(\beta)} \equiv \begin{Bmatrix}{z_{C} - {\int_{0}^{S_{C}}{\sin \quad {\theta (s)}\quad {s}}}} \\{\theta_{C} - \theta_{C\quad D}} \\ {E\quad I\frac{\theta}{s}} \middle| {}_{E}{{{+ F_{X\quad E}}R_{S}\sin \quad \theta_{E}} - {F_{ZE}R_{S}\cos \quad \theta_{E}}} \end{Bmatrix}},$

[0043] and wherein F_(X)(s) and F_(Z)(s) are functions of s describingthe internal x-directed and z-directed forces acting on the stamp'scross section at s per unit depth of the stamp in the y direction,F_(XE)≡F_(X)(s_(E)),F_(ZE)≡F_(Z)(s_(E)), β is a vector of parametersthat are unknown a priori, ${\beta = \begin{Bmatrix}s_{0} \\F_{X0} \\F_{Z0}\end{Bmatrix}},$

[0044] s₀ is the aforementioned arc-length coordinate of the contactfront, F_(X0)≡F_(X)(s₀), F_(Z0)≡F_(Z)(s₀), E is Young's modulus of thestamp, I is the area moment of inertia of the stamp's cross section perunit depth in the y-direction, w is the weight per unit area of thestamp, p(s) and f(s) are functions of s describing forces applied normalto the stamp and tangential to the stamp respectively by theprint-force-application system, the stamp-control system and the printsurface, s_(C) is the value of arc-length coordinate s at point C,θ_(C)≡θ(s_(C)) is the angle of the arc at point C, and θ_(CD) is theaforementioned angle of the stamp-control system's contact surface; and

[0045] wherein for each configuration the solution for x_(P) and z_(P)is derived by means of the “shooting method”, whereby an initial valueβ⁽⁰⁾ of β is guessed, the differential equations are solved to yieldT(β⁽⁰⁾) and$\lbrack \frac{\partial T}{\partial\beta} \rbrack_{\beta = \beta^{(0)}}^{- 1},$

[0046]  Newton-Raphson iteration$\beta^{({n + 1})} = {\beta^{(n)} - {\lbrack \frac{\partial T}{\partial\beta} \rbrack_{\beta = \beta^{(n)}}^{- 1}{T( \beta^{(n)} )}}}$

[0047]  is applied to obtain a refined vector β⁽¹⁾, whereupon thedifferential equations are solved again; this iteration procedure beingapplied repeatedly until the correct auxiliary boundary conditionsT(β)=0 are achieved to within some tolerance.

[0048] In another aspect of the printer, the trajectory comprises aplurality of configurations, each configuration described by thecoordinate s₀ x₀ of the contact front and by corresponding coordinatesx_(P), z_(P) of the pivot line given by the equations:

x _(P) =x _(E) +R _(s) cos θ_(E)

z _(P) =z _(E) +R _(s) sin θ_(E),

[0049] where x_(E)=∫₀ ^(L) cos θ(s)ds and z_(E)=∫₀ ^(L) sin θ(s)ds, andwhere the mathematical function θ(s) describing the shape of the arc fora given configuration is assumed to be

θ(s)=0 for 0≦s≦s ₀,

[0050] whereas for s>s₀, θ(s) is determined in stamp segments OC and DEby solution of the differential equations ${\frac{u}{s} = {F(u)}},$

[0051] the lower-end boundary conditions ${{u_{0} \equiv \begin{Bmatrix}u_{10} \\u_{20}\end{Bmatrix} \equiv \begin{Bmatrix}\theta_{0} \\ \frac{\theta}{s} |_{0}\end{Bmatrix}} = \begin{Bmatrix}0 \\\kappa_{0}\end{Bmatrix}},$

[0052] and the upper-end boundary condition${ {{T(\beta)} \equiv {{EI}\frac{\theta}{s}}} \middle| {}_{E}{{{+ F_{XE}}R_{S}\sin \quad \theta_{E}} - {F_{ZE}R_{S}\cos \quad \theta_{E}}}  = 0},{wherein}$${u \equiv \begin{Bmatrix}u_{1} \\u_{2}\end{Bmatrix} \equiv \begin{Bmatrix}\theta \\\frac{\theta}{s}\end{Bmatrix}},{{F(u)} \equiv \{ \begin{matrix}u_{2} \\{{\frac{F_{X}(s)}{EI}\sin \quad u_{1}} - {\frac{F_{Z}(s)}{EI}\cos \quad u}}\end{matrix}_{1} \}},$

[0053] κ₀ is a specified curvature at point O, E is Young's modulus ofthe stamp, I is the area moment of inertia of the stamp's cross sectionper unit depth in the y-direction, w is the weight per unit area of thestamp, F_(x)(s) and F_(z)(s) are the x-directed and z-directed stampforces per unit length of stamp in the y direction, given by${F_{x}(s)} = \{ {{\begin{matrix}{{F_{x0},}} & {0 \leq s \leq s_{C}} \\{{{F_{x0} + {\Delta \quad F_{x}}},}} & {{s_{D} \leq s \leq s_{E}},}\end{matrix}{and}{F_{z}(s)}} = \{ \begin{matrix}{{0,}} & {0 \leq s \leq s_{0}} \\{{{w( {s - s_{0}} )},}} & {s_{0} \leq s \leq s_{C}} \\{{{{w( {s - s_{0}} )} + {\Delta \quad F_{z}}},}} & {{s_{D} \leq s \leq s_{E}},}\end{matrix} } $

[0054] in which F_(x0)≡F_(x)(s₀)≡β is a parameter that is unknown apriori, and the differences ΔF_(x) and ΔF_(z) are respectively thedifferences

ΔF _(x) ≡F _(x)(s _(D))−F _(x)(s _(C))

ΔF _(z) ≡F _(z)(s _(D))−F _(z)(s _(C))

[0055] that occur across stamp segment CD where the stamp-control systemcontacts the stamp, the values of which differences, along with thevalue of the difference$ {{\Delta \quad \kappa} \equiv \frac{\theta}{s}} \middle| {}_{D}{- \frac{\theta}{s}} |_{C},$

[0056] may be calculated from the three equations of static equilibriumfor the stamp under the action of forces applied to the stamp by thestamp-control system, these three differences together with θ_(D)=θ_(C)permitting numerical integration for stamp segment DE to proceedimmediately from the numerical-integration result obtained at the finalpoint C in stamp segment OC, and wherein for each configuration thesolution for x_(P) and z_(P) is derived by means of the “shootingmethod”, whereby an initial value β⁽⁰⁾ of β is guessed, the differentialequations are solved to yield T(β⁽⁰⁾) and$\lbrack \frac{\partial T}{\partial\beta} \rbrack_{\beta = \beta^{(0)}}^{- 1},$

[0057] Newton iteration$\beta^{({n + 1})} = {\beta^{(n)} - {\lbrack \frac{\partial T}{\partial\beta} \rbrack_{\beta = \beta^{(n)}}^{- 1}{T( \beta^{(n)} )}}}$

[0058] is applied to obtain a refined vector β⁽¹⁾, whereupon thedifferential equations are solved again; this iteration procedure beingapplied repeatedly until the correct auxiliary boundary conditionsT(β)=0 are achieved to within some tolerance.

[0059] In another aspect of the printer, a stamp-control system movablealong the x-axis, wherein, throughout the trajectory, each xz positionof the upper stamp clamp is a function of the displacement x_(C) of thestamp-control system along the x-axis; the trajectory being effective inlaying the stamp down upon the print surface such that the stamp is incontinuous contact with a contact surface of the stamp-control system,the location of the contact surface being characterized by an arc-lengthcoordinate s_(C) that increases as the trajectory progresses.

[0060] In another aspect of the printer, the stamp-control system isdisposed along a line_(C) parallel to the y-axis, line_(C) intersectingthe trajectory plane at point C having coordinates x_(C) and z_(C),where z_(C) is a fixed, positive z-coordinate during any one printingoperation, whereas x_(C) increases as the trajectory progresses, incoordination with the contact-front coordinate x₀.

[0061] In another aspect of the printer the contact surface of thestamp-control system is a plane delimited in the x direction by twolines _(C) and _(D) separated by a fixed distance W_(CD), these linesbeing parallel to the y-axis and intersecting the trajectory plane atpoints C and D respectively, these points having coordinates(x_(C),z_(C)) and (x_(D),z_(D)) respectively, such that the contactsurface is defined by the three parameters (x_(C),z_(C),θ_(CD)), where$\theta_{CD} \equiv {\tan^{- 1}( \frac{z_{D} - z_{C}}{x_{D} - x_{C}} )}$

[0062] is the angle between the contact surface and the print plane, andsuch that the stamp angle θ(s) between arc-length coordinates s=s_(C)and s=S_(D) is substantially equal to θ_(CD); that is,

θ(s)≈θ_(CD) for s _(C) ≦s ≦s _(D).

[0063] In another aspect of the printer, the upper stamp clamp ispivoted about a pivot line_(P) parallel to the y axis and intersectingthe xz plane at point P having coordinates x_(P) and z_(P); the stampattaching to the upper stamp clamp along an upper-clamp line_(E)parallel to the y axis and intersecting the xz plane at point E havingcoordinates x_(E) and z_(E); the upper-clamp line_(E) being disposed onthe upper stamp clamp at a radius R_(s) from the pivot line_(P), suchthat the total arc length s_(E) from the lower stamp clamp to theline_(E) is s_(E) L, where L is the known, free length of the stamp; andwherein the stamp attaches to the upper-clamp line_(E) at an angleθ_(E)≡θ(L).

[0064] In another aspect of the printer, the trajectory comprises aplurality of configurations, each configuration described by thecoordinate s₀ x₀ of the contact front and by corresponding coordinatesx_(P), z_(P) of the pivot line given by the equations:

x _(P) =x _(E) +R _(s) cos θ_(E)

z _(P) =z _(E) +R _(s) sin θ_(E),

[0065] where x_(E)=∫₀ ^(L) cos θ(s)ds and z_(E)=∫₀ ^(L) sin θ(s)ds, andwhere the mathematical function θ(s) describing the shape of the arc fora given configuration is assumed to be

θ(s)=0 for 0≦s≦s ₀,

[0066] whereas for s>s₀, θ(s) is determined by solution of thedifferential equations ${\frac{u}{s} = {F(u)}},$

[0067] the lower-end boundary conditions ${{u_{0} \equiv \begin{Bmatrix}u_{10} \\u_{20} \\u_{30} \\u_{40}\end{Bmatrix} \equiv \begin{Bmatrix}\theta_{0} \\ \frac{\theta}{s} |_{0} \\F_{X_{0}} \\F_{Z_{0}}\end{Bmatrix}} = \begin{Bmatrix}0 \\0 \\F_{X_{0}} \\F_{Z_{0}}\end{Bmatrix}},$

[0068] and the auxiliary boundary conditions

T(β)=0,

[0069] wherein ${u \equiv \begin{Bmatrix}u_{1} \\u_{2} \\u_{3} \\u_{4}\end{Bmatrix} \equiv \begin{Bmatrix}\theta \\\frac{\theta}{s} \\{F_{x}(s)} \\{F_{z}(s)}\end{Bmatrix}},{{F(u)} \equiv \begin{Bmatrix}u_{2} \\{{\frac{u_{3}}{EI}\sin \quad u_{1}} - {\frac{u_{4}}{EI}\cos \quad u_{1}}} \\{{{- {p(s)}}\sin \quad u_{1}} - {{f(s)}\cos \quad u_{1}}} \\{w + {{p(s)}\cos \quad u_{1}} - {{f(s)}\sin \quad u_{1}}}\end{Bmatrix}},{{T(\beta)} \equiv \begin{Bmatrix}{z_{C} - {\int_{0}^{s_{C}}{\sin \quad {\theta (s)}{s}}}} \\{\theta_{C} - \theta_{CD}} \\ {{EI}\frac{\theta}{s}} \middle| {}_{E}{{{+ F_{XE}}R_{S}\sin \quad \theta_{E}} - {F_{ZE}R_{S}\cos \quad \theta_{E}}} \end{Bmatrix}},$

[0070] and wherein F_(X)(s) and F_(Z)(s) are functions of s describingthe internal x-directed and z-directed forces acting on the stamp'scross section at s per unit depth of the stamp in the y direction,F_(XE)≡F_(X)(s_(E)), F_(ZE)≡F_(Z)(s_(E)), β is a vector of parametersthat are unknown a priori, ${\beta = \begin{Bmatrix}s_{0} \\F_{X_{0}} \\F_{Z_{0}}\end{Bmatrix}},$

[0071]₀ is the aforementioned arc-length coordinate of the contactfront, F_(X0)≡F_(X)(0), F_(Z0)≡F_(Z)(0), E is Young's modulus of thestamp, I is the area moment of inertia of the stamp's cross section perunit depth in the y-direction, w is the weight per unit area of thestamp, p(s) and f(s) are functions of s describing forces applied normalto the stamp and tangential to the stamp respectively by theprint-force-application system, the stamp-control system and the printsurface, s_(C) is the value of arc-length coordinate s at point C,θ_(C)≡θ(s_(C)) is the angle of the arc at point C, and θ_(CD) is theaforementioned angle of the stamp-control system's contact surface; and

[0072] wherein for each configuration the solution for x_(P) and z_(P)is derived by means of the “shooting method”, whereby an initial valueβ⁽⁰⁾ of β is guessed, the differential equations are solved to yieldT(β⁽⁰⁾) and$\lbrack \frac{\partial T}{\partial\beta} \rbrack_{\beta = \beta^{(0)}}^{- 1},$

[0073]  Newton-Raphson iteration$\beta^{({n + 1})} = {\beta^{(n)} - {\lbrack \frac{\partial T}{\partial\beta} \rbrack_{\beta = \beta^{(n)}}^{- 1}{T( \beta^{(n)} )}}}$

[0074]  is applied to obtain a refined vector β⁽¹⁾, whereupon thedifferential equations are solved again; this iteration procedure beingapplied repeatedly until the correct auxiliary boundary conditionsT(β)=0 are achieved to within some tolerance.

[0075] In another aspect of the printer, the trajectory comprises aplurality of configurations, each configuration described by thecoordinate s_(B) of the approximate contact front and by correspondingcoordinates x_(P), z_(P) of the pivot line given by the equations

x _(P) =x _(E) +R _(s) cos θ_(E)

z _(P) =z _(E) +R _(s) sin θ_(E),

[0076] where x_(E)=∫₀ ^(L) cos θ(s)ds and z_(E)=∫₀ ^(L) sin θ(s)ds, andwhere the mathematical function θ(s) describing the shape of the arc fora given configuration is assumed to be

θ(s)=0 for 0≦s≦s _(B),

[0077] whereas for s>s_(B), θ(s) is determined in stamp segments OC andDE by solution of the differential equations${\frac{u}{s} = {F(u)}},$

[0078] the lower-end boundary conditions ${{u_{B} \equiv \begin{Bmatrix}u_{1B} \\u_{2B}\end{Bmatrix} \equiv \begin{Bmatrix}\theta_{B} \\ \frac{\theta}{s} |_{B}\end{Bmatrix}} = \begin{Bmatrix}0 \\\kappa_{B}\end{Bmatrix}},$

[0079] and the upper-end boundary condition${ {{T(\beta)} \equiv {{EI}\frac{\theta}{s}}} \middle| {}_{E}{{{+ F_{XE}}R_{S}\sin \quad \theta_{E}} - {F_{ZE}R_{S}\cos \quad \theta_{E}}}  = 0},{wherein}$${u \equiv \begin{Bmatrix}u_{1} \\u_{2}\end{Bmatrix} \equiv \begin{Bmatrix}\theta \\\frac{\theta}{s}\end{Bmatrix}},{{F(u)} \equiv \begin{Bmatrix}u_{2} \\{{\frac{F_{x}(s)}{EI}\sin \quad u_{1}} - {\frac{F_{z}(s)}{EI}\cos \quad u_{1}}}\end{Bmatrix}},$

[0080] κ_(B) is a specified curvature at point B, E is Young's modulusof the stamp, I is the area moment of inertia of the stamp's crosssection per unit depth in the y-direction, w is the weight per unit areaof the stamp, F_(x)(s) and F_(z)(s) are the x-directed and z-directedstamp forces per unit length of stamp in they direction, given by${F_{x}(s)} = \{ {{\begin{matrix}{{F_{x0},}} & {0 \leq s \leq s_{C}} \\{{{F_{x0} + {\Delta \quad F_{x}}},}} & {{s_{D} \leq s \leq s_{E}},}\end{matrix}{and}{F_{z}(s)}} = \{ \begin{matrix}{{0,}} & {0 \leq s \leq s_{0}} \\{{{w( {s - s_{0}} )},}} & {s_{0} \leq s \leq s_{C}} \\{{{{w( {s - s_{0}} )} + {\Delta \quad F_{z}}},}} & {{s_{D} \leq s \leq s_{E}},}\end{matrix} } $

[0081] in which F_(x0)≡F_(x)(s₀)≡β is a parameter that is unknown apriori, and the differences ΔF_(x) and ΔF_(z) are respectively thedifferences

ΔF _(x) ≡F _(x)(s _(D))−F _(x)(s _(C))

ΔF _(z) ≡F _(z)(s _(D))−F _(z)(s _(C))

[0082] that occur across stamp segment CD where the stamp-control systemcontacts the stamp, the values of which differences, along with thevalue of the difference$ {{\Delta \quad \kappa} \equiv \frac{\theta}{s}} \middle| {}_{D}{- \frac{\theta}{s}} |_{C},$

[0083] may be calculated from the three equations of static equilibriumfor the stamp under the action of forces applied to the stamp by thestamp-control system, these three differences together with θ_(D)=θ_(C)permitting numerical integration for stamp segment DE to proceedimmediately from the numerical-integration result obtained at the finalpoint C in stamp segment OC, and wherein for each configuration thesolution for x_(P) and z_(P) is derived by means of the “shootingmethod”, whereby an initial value β⁽⁰⁾ of β is guessed, the differentialequations are solved to yield T(β⁽⁰⁾) and$\lbrack \frac{\partial T}{\partial\beta} \rbrack_{\beta = \beta^{(0)}}^{- 1},$

[0084] Newton iteration$\beta^{({n + 1})} = {\beta^{(n)} - {\lbrack \frac{\partial T}{\partial\beta} \rbrack_{\beta = \beta^{(n)}}^{- 1}{T( \beta^{(n)} )}}}$

[0085] is applied to obtain a refined vector β⁽¹⁾, whereupon thedifferential equations are solved again; this iteration procedure beingapplied repeatedly until the correct auxiliary boundary conditionsT(β)=0 are achieved to within some tolerance.

[0086] In another aspect of the printer, the print-force-applicationsystem comprises a flat-iron.

[0087] In another aspect of the printer, the stamp-control systemcomprises a vacuum bar.

[0088] Disclosed is a printing apparatus, comprising a receiver meanswhose receiving surface lies in an xy plane, the normal to the surfacedefining a z-axis direction, a lower stamp clamp means for fixing afirst edge of a stamp, an upper stamp clamp means for holding a secondedge of a stamp for movement in the xz directions, a flexible stampmeans for printing to the receiver, said flexible stamp in substantiallythe form of a sheet defining edges, the first edge of which is affixedto the lower stamp clamp, and the opposing second edge of which isaffixed to the upper stamp clamp, thereby allowing the stamp to hang ina curve under gravity and the sheet's own stiffness, such that everynormal to the stamp's curved surface lies substantially parallel to thexz plane, and a trajectory-producing means for moving the upper stampclamp along a prescribed trajectory in the xz plane, such that the stampis draped upon the receiving surface in a manner that causes thecurvature of the stamp near a contact front at a point B to be constantthroughout the trajectory.

[0089] Another aspect of the apparatus further comprises print-forceapplication means for applying pressure upon the stamp means against thereceiver means and for defining the contact front.

[0090] Another aspect of the apparatus further comprises stamp-controlmeans for defining a point C through which the curvature of the sheetwill pass throughout the trajectory.

BRIEF DESCRIPTION OF THE DRAWINGS

[0091]FIG. 1 is a schematic drawing of a micro-contact printing process.

[0092]FIG. 2 is a schematic drawing of a two-layer stamp of the typeused by this invention.

[0093]FIG. 3 is a block diagram specifying the components of a printingmachine.

[0094]FIG. 4 is a three-dimensional view of an embodiment of theprinting machine of the invention, prior to loading the stamp.

[0095]FIG. 5 is a three-dimensional view of an embodiment of theprinting machine of the invention, after loading the stamp.

[0096]FIGS. 6a, 6 b, and 6 c are three-dimensional views of the printingmachine of the invention at the start, the middle, and the end of theprinting process, respectively.

[0097]FIG. 7 is a three-dimensional view of a base table.

[0098]FIG. 8a is a three-dimensional view of a print table before areceiver is loaded.

[0099]FIGS. 8b and 8 c are orthographic projections of the print table.

[0100]FIG. 9 is a three-dimensional view of the print table after thereceiver is loaded.

[0101]FIG. 10 is a three-dimensional view of a lower stamp clamp.

[0102]FIG. 11 is cross-sectional diagram of the print table, the lowerstamp clamp, the stamp and the receiver.

[0103]FIG. 12 is a three-dimensional view of the bottom surface of avacuum plate that is part of the lower stamp clamp.

[0104]FIG. 13 is a close-up of an adjustment mechanism for the lowerstamp clamp.

[0105]FIG. 14 is a close-up of another adjustment mechanism for thelower stamp clamp.

[0106]FIG. 15 is a cross-sectional diagram of screws that locate andattach the stamp to the lower stamp clamp.

[0107]FIG. 16 is a three-dimensional view of the upper stamp clamp as itappears when mounted in the printing machine.

[0108]FIG. 17 is a cross-sectional diagram of the upper stamp clamp.

[0109]FIG. 18 is a three-dimensional view of the upper stamp clamp as itappears, remote from the printer, while attaching a stamp.

[0110]FIG. 19 is a three-dimensional view of a carriage.

[0111]FIG. 20 is another three-dimensional view of the carriage.

[0112]FIG. 21 is yet another three-dimensional view of the carriage.

[0113]FIG. 22 is a schematic diagram of the system, showing rationalefor a flat-iron and a vacuum bar.

[0114]FIG. 23a and 23 b are, respectively, a three-dimensional and anorthographic view of the flat-iron.

[0115]FIG. 24a is a cross-sectional schematic diagram of the flat-iron;FIG. 24b shows the pressure profile in the air bearing.

[0116]FIG. 25 is a three-dimensional view of the vacuum-bar assembly.

[0117]FIG. 26 is a three-dimensional close-up view of the front end ofthe vacuum-bar assembly.

[0118]FIG. 27 is a three-dimensional close-up view of the rear end ofthe vacuum-bar assembly.

[0119]FIG. 28 shows various orthographic projections of the rear end ofthe vacuum-bar assembly.

[0120]FIG. 29 shows orthographic projections of the vacuum bar.

[0121]FIG. 30 is a schematic diagram showing the various stamp segmentsused in the mathematical analysis of the invention.

[0122]FIG. 31 is a free-body diagram of a differential element of thestamp.

[0123]FIG. 32 is a diagram showing forces and dimensions on the lowersurface of the vacuum bar.

[0124]FIG. 33 is a close-up diagram of stamp segment AB.

[0125]FIG. 34 is a close-up diagram of stamp segment CD.

[0126]FIG. 35 is a close-up diagram illustrating the stamp's top-endboundary condition.

[0127]FIG. 36 shows various mathematically generated solutions for theshape of a stamp.

[0128]FIG. 37 is a schematic diagram explaining the graphicalmethodology used in FIGS. 38 through 42.

[0129]FIGS. 38 through 45 are various measurements of feature placementerrors obtained with a prototype of the invention.

[0130]FIG. 46 shows the meaning of forward and reverse printingdirections.

DETAILED DESCRIPTION OF THE INVENTION 1. Overall Description

[0131]FIG. 1 shows a schematic of the printing process accomplished bythe invention. Apart from the printing machine itself, the printingprocess requires three elements, as shown in FIG. 1a: a stamp 5 having aprinting surface 10 embossed with a raised pattern 15; ink 20 that coatsthe raised pattern 15; and a receiver 25 to whose receiver surface 30the ink 20 will be transferred during the printing process by mechanicalcontact of raised pattern 15 and receiver surface 30, as shown in FIG.1b. The printing machine (not shown in FIG. 1) must accomplish thistransfer in such a way that after separation of the stamp 5 from thereceiver 25 (FIG. 1c), the placement of transferred ink on the receiversurface 30 faithfully replicates the raised pattern 15. The inventionprovides such a printing machine having a number of advantages overprior-art machines.

[0132] Although the printing machine may be used with any type of stampconsistent with the following detailed description, the preferredembodiment of the machine uses the type of stamp described in commonlyassigned Biebuyck et al., U.S. Pat. No. 5,817,242, entitled STAMP FORLITHOGRAPHIC PROCESS, issued Oct. 6, 1998, the disclosures of which areincorporated by reference herein in their entirety. As shown in FIG. 2,this type of stamp comprises two bonded layers, a polymer layer 35, uponwhose printing surface 10 the raised pattern 15 is embossed, and aback-plane layer 40, typically composed of metal or other materialhaving a high modulus of elasticity. The back-plane layer 40 provideshigh lateral stiffness, and therefore high feature-placement accuracy ofthe printed pattern on receiver surface 30. Such accuracy is difficultto achieve—particularly for large-area stamps—using the low-moduluspolymer layer 35 alone. As drawn in FIG. 2, the area of the back-planelayer 40 is typically somewhat larger than that of the polymer layer 35,for reasons explained subsequently.

[0133] Referring to FIG. 3, the printing machine 100 described in thisinvention will preferably comprise six subsystems, including abase-table assembly 105, a print-table assembly 110, a linear-motionsystem 115, a lower stamp clamp 120, a carriage 125, and an upper stampclamp 130.

[0134] As also shown in FIG. 3, four of these six subsystems may befurther divided into more detailed subsystems. The base-table assembly105 comprises a horizontal base table 135, a vertical base table 140,and a vibration-control system 145. The print-table assembly 110comprises the print-table itself, 150, as well as pneumatics 155. Thelinear-motion system 115 comprises three axes of motion (160, 180, and200), requiring three sets of linear-motion stages (165, 185, 205);motors (170, 190, and 210) and motor drivers (175, 195, 215); amulti-axis motor controller 220; a computer 225 utilizing standardhardware 230 but specialized software 235; and a joystick 240 or similarinput device. The lower stamp clamp 120 comprises a vacuum chuckassembly 245 that may be micro-positioned with respect to the printtable 150, as well as pneumatics 250 to operate the vacuum chuck. Thecarriage 125 comprises a mechanical frame 255; a print-force-applicationsystem 260 for applying the force of contact between the stamp 5 and thereceiver 25 during printing; and a stamp-control system 265 to insurethat the critical geometry of the stamp is maintained throughout theprinting process, thereby to achieve the best possible printed replicaof the embossed pattern 15 on the receiver 25.

[0135] The latter two systems may be further subdivided into components.The print-force-application system 260 comprises anair-bearing-supported flat-iron 270 to apply the force of contactbetween stamp 5 and receiver 25 during printing, and pneumatics 275 tocontrol the flow of air to the flat-iron. The stamp-control systemcomprises a vacuum-bar assembly 280 to control the geometry of the stampduring printing, and pneumatics 285 to operate the vacuum bar.

[0136]FIG. 4 depicts the entire printing machine 100 without the stampinstalled. For reference, an xyz coordinate system 290 is shown; onsubsequent figures, the xyz orientation of this coordinate system (butnot the origin) remains consistent. The base-table assembly 105 (fromFIG. 3) comprises a horizontal base table 135 and a vertical base table140, affixed to each other at right angles, forming an “L” shape. Theprint table 150 rests on the horizontal base table 135, and the lowerstamp clamp 120 rests on the print table's stepped-down surface 295,which is preferably machined lower than its elevated receiver surface300, for reasons clarified later. Pneumatic controls represented by 155,250, 275, and 285 are shown at the right end of the horizontal basetable 135.

[0137] The system of three computer-driven stages (165, 185, 205) areaffixed to the vertical base table 140. Two of these stages, 165 and185, which correspond to axes (x₁,z), form a two-axis set: horizontalstage 165 is affixed to the vertical base table 140; stage 185 isaffixed to faceplate 305 of stage 165. Thus faceplate 310 of z stage 185is capable of executing two-dimensional motion parallel to the xz plane.The third computer-driven stage, 205, corresponding to axis x₂, isaffixed to the vertical base table, parallel to stage 165 and below it,in such a way that the z stage 185 can pass over the body of the x₂stage 205 without interference.

[0138] The upper stamp clamp 130 is affixed to faceplate 310 of z-stage185; thus, the upper stamp clamp 130 can move in two dimensions parallelto the xz plane. Likewise, the carriage 125 is affixed to faceplate 315of x₂-stage 205; thus, the carriage can move in one dimension parallelto the x-axis. Because the motors (170, 190, 210) driving all threestages are computer controlled via the same controller 220 (from FIG.3), it is possible to provide coordinated triple-axis movement of theupper stamp clamp and the carriage simultaneously.

[0139] Referring to FIGS. 5 and 6, the receiver 25 is held fixed,throughout the printing process, to the print table's elevated receiversurface 300. As shown in FIG. 6b, one end of the stamp's back-planelayer 40, hereafter called the stationary end 320, is attached to thelower stamp clamp 120, such that the stamp's polymer layer 35 facesdownward. This stationary end 320 remains fixed during printing. Theopposite end of the stamp's back-plane layer 40, hereafter called themovable end 325, is attached to the upper stamp clamp 130. Because thestamp is flexible, when it is mounted in the printing machine, suspendedbetween the upper stamp clamp 130 and the lower stamp clamp 120, ithangs in a natural curve, as shown.

[0140]FIG. 6 shows how the printing process proceeds. Specifically,FIGS. 6a, 6 b, and 6 c show the beginning, middle, and end of theprinting process respectively. As shown by comparison of the threefigures, the upper stamp clamp 130 moves in two dimensions (xz) duringprinting—downward and to the right, thereby laying the stamp's polymerlayer 35 gradually upon the receiver 25. Simultaneously, in a motioncoordinated with that of the upper stamp clamp 130, the carriage 125moves to the right, so that the systems it carries—theprint-force-application system 255 and the stamp-control system 260—canapply the contact force and geometrical control necessary to achieveuniform, accurate printing. When the end of the printing process isreached (i.e., when the stamp is fully in contact with the receiver), orperhaps after an intervening delay, the stamp is peeled from thereceiver by reversing the three-axis motions used for printing. Whenpeeling is done—when the stamp is completely separated from the receiver25—the receiver, now patterned with the ink 20, may be removed from theprinting machine.

2. Detailed Description of Sub-Systems

[0141] 2.1 Base-Table Assembly

[0142]FIG. 7 shows the base-table assembly 105 (from FIG. 3) in moredetail. In addition to the horizontal base table 135 and the verticalbase table 140, it preferably comprises two diagonal struts 330 or othersuitable reinforcing means, to hold the vertical base tableperpendicular to the horizontal base table. These may be attached by sixstrut blocks 335 for attachment of the diagonal struts 330 to the basetables 135 and 140 as well as for attachment of the tables to eachother. Also provided is a vibration-control system 145. In a prototypeof the invention constructed to generate the print images of FIGS. 38through 45, the horizontal base table 135, vertical base table 140, andvibration-isolation system 145 were purchased from Newport Corporation(Fountain Valley, Calif.). The thickness of the tables (135, 140) andload-bearing capacity of the vibration-control system 145 depend on thex and y dimensions of the receiver 25 (from FIG. 3). In the prototype ofthe invention, the receiver dimensions are 381 mm×381 mm (15″×15″). Forthis case the horizontal base table 135 is 914 mm×1,524 mm×102 mm thick(36″×60″×4″ thick), the vertical base table 140 is 914 mm×1,524 mm×51 mmthick (36″×60″×2″ thick), and the vibration-isolation system 145 has aload capacity of 5,780 N (1300 lbs).

[0143] 2.2 Print-Table Assembly

[0144]FIGS. 8a, 8 b, and 8 c show an embodiment of the print table 150in more detail. FIG. 8a is a three-dimensional view of the print table150 with a lower stamp clamp 120 mounted on it. FIGS. 8b and 8 c arerespectively a top view and a rear view of the print table 150 alone. Asshown in FIG. 8a, a preferred top surface of the print table 150comprises a step 340, which divides the surface into an elevatedreceiver surface 300 and, parallel to it, the stepped-down surface 295.Preferably, both of these surfaces will be flat to a high degree ofprecision. Stepped-down surface 295 may have threaded holes 345 (FIG.8b) or other mounting means as will be described below in conjunctionwith the lower stamp clamp 120. Elevated receiver surface 300 willpreferably have a connected series of narrow, shallow grooves 350 orother suitable vacuum carrying means, which provide a vacuum chuck tohold receiver 25 fixed to elevated receiver surface 300 during theprinting process, and also provide an air-pressure chuck to break thevacuum seal when the printing process is finished. The grooves 350 inelevated receiver surface 300 are alternately exposed to vacuum and airpressure, during and after the printing process respectively, by meansof one or more vacuum openings 355, drilled perpendicular to receiversurface 300, and which communicate with grooves 350. The vacuum openings355 may intersect a pressure-supply hole 360 (FIG. 8c) drilled at rightangles to it, parallel to receiver surface 300, from the rear surface365 of print table 150. Where pressure-supply hole 360 exits print table150 at rear surface 365, it is terminated with a common pneumaticfitting 370, which is connected to pneumatic controls 155, of a typewell known in the art, that can alternately supply vacuum or airpressure, as described above, to grooves 350.

[0145]FIG. 9 shows that, before receiver 25 is affixed to elevatedreceiver surface 300 using vacuum pressure applied to grooves 350, thereceiver will preferably be located precisely on the surface byalignment means 375 such as, for example, a plurality of banking pinsthreaded into the three threaded holes 380 shown on FIG. 8b. The bankingpins 375 protrude above receiver surface 300 by an amount just slightlyless than the thickness of the receiver 25. Receiver 25 is banked to thecylindrical surfaces of banking pins 375 by application of lateralforces 385 applied in the -x and -y directions. The banking pins 375 arelocated such that, when the receiver 25 is registered against the pins'side surfaces, the receiver's leading edge 390 is substantiallyperpendicular to the line formed by the print table's step 340.

[0146] Preferably, the print table 150 is made of a material, such asgranite, whose surfaces may be made very flat. Custom block ofgranite—comprising step 340, threaded holes 345, grooves 350, vacuumhole 355, pressure-supply hole 360, pneumatic fitting 370, and threadedholes 380—may be ordered from various commercial sources, such as L. S.Starrett Co., Granite Surface Plate Division, Mt. Airy, N.C. Solidgranite is a preferred material for this application because theflatness of the receiver surface 300, upon which receiver 25 restsduring printing, significantly influences the accuracy of the transferof the pattern 15 from the stamp 5 to the receiver 25. As is well-knownin the art of tool-making, a lapped granite surface provides a durable,high-precision flat surface.

[0147] 2.3 Lower Stamp Clamp

[0148]FIG. 10 shows the vacuum-chuck assembly 245 (from FIG. 3) of lowerstamp clamp 120 (from FIG. 3). This assembly, which rests uponstepped-down surface 295 of print table 150, comprises a vacuum plate395; a stamp attachment means 400 such as, for example, one or morelocating screws that mate with threaded locating holes 405 (shown inFIG. 9) for attachment of the stamp to the vacuum plate 395; and threemicrometer assemblies 425, 430, and 435. In the embodiment shown in thedrawings, a first micrometer assembly 425 is affixed to front surface440 of vacuum plate 395. Likewise, a second micrometer assembly 430—amirror image of 425—is affixed to the rear surface of vacuum plate 395.Yet a third micrometer assembly 435 is affixed to the stepped-downsurface 295 of print table 150 using two threaded holes 345. The threemicrometer assemblies 425, 430, and 435 are used to adjust the positionand orientation of the vacuum plate 395 with respect to the print table150.

[0149] As shown in FIG. 11, the vacuum plate 395 is machined tothickness h₃₉₅ satisfying the following equation:

h ₃₉₅ =h ₃₄₀ +h ₂₅ +h ₃₅,

[0150] where h₃₄₀ is the height of step 340, h₂₅ is the thickness ofreceiver 25, and h₃₅ is the thickness of the stamp's polymer layer 35.In this way, when the polymer layer 35 is brought into contact with thereceiver during the printing process, the stamp's back-plane layer 40lies flat, as shown in FIG. 11.

[0151] As further shown in FIG. 11, and also in FIG. 12 (which is abottom view of vacuum plate 395), the bottom surface 445 of vacuum plate395 is machined with a number of shallow grooves 450 or other suitablevacuum carrying means. These grooves provide a vacuum chuck to holdvacuum plate 395 fixed to surface 295 during the printing process, butalso provide, as needed during machine setup, an air-pressure chuck bymeans of which the vacuum plate 395 is floated on an air cushion tofacilitate translational and rotational adjustment of vacuum plate 395with respect to the print table 150, as discussed further below inconnection with the micrometer assemblies 425, 430, and 435. The grooves455 in surface 450 may be alternately exposed to vacuum and airpressure, as needed, by means of a bleed hole 455 drilled perpendicularto surface 445 and communicates with grooves 450. The bleed hole 455intersects a pressure-supply hole 460 drilled at right angles to it,parallel to surface 445, as shown in FIG. 11. The pressure-supply hole460 is terminated with a thread suitable for a pneumatic fitting 465.Pneumatic controls 250, of a type well know in the art, are connected tothe fitting 465 to alternately supply the vacuum and air pressure asdescribed above.

[0152]FIG. 13 is a close-up of first micrometer assembly 425, whichcomprises a micrometer head 470, and a mounting means, such as anL-bracket 475, and screws 480 that affix L-bracket 475 to front surface440 of vacuum plate 395. The body of the micrometer head 470 is affixedto the L-bracket 475 using, for example, a slotted-hole clamparrangement, wherein a clamping screw 485 forces together the legs 490on either side of the slot 495, thereby gripping the micrometer head475, such that the axis of the micrometer head is parallel to frontsurface 440 of vacuum plate 395, and rigidly connected to it.Non-rotating spindle 500 of micrometer head 475, outfitted withball-bearing tip 505, is extended so that the ball-bearing tip 505 bearsagainst step 340 of print table 150, such that, by further extension ofthe spindle 500, the distance 510 (between step 340 and vacuum plate 395near the front of the print table 150) is increased. Conversely,distance 510 may be decreased by retraction of the spindle, providedonly that a force (manual, spring-loaded, etc.) is applied to vacuumplate 395, in the positive x direction, to keep the ball-bearing tip 505and the step 340 in intimate contact. The ball-bearing tip assures arolling, single-point contact between the micrometer and the step 340.

[0153] As shown in FIG. 14, there is preferably provided a secondmicrometer assembly 430, affixed to the rear surface of vacuum plate395, a mirror image of first micrometer assembly 425. Thus, by analogyto the previous paragraph, second micrometer assembly 430 is used tomodulate the distance 515 between step 340 and vacuum plate 395 near therear of the print table 150. Thus, the two micrometer assemblies 425 and430, by modulating distances 510 and 511 respectively, move vacuum plate395 in the two degrees of freedom x and θ.

[0154] As further shown in FIG. 14, there is preferably provided a thirdmicrometer assembly 435 comprising a micrometer head 520, a straightbracket 525, and screws 530 or similar means to affix straight bracket525 or other mount means to the stepped-down surface 295 of the printtable 150. The body of the micrometer head 520 may be affixed to thestraight bracket 525, using the slotted-hole clamp arrangement describedin connection with FIG. 13, such that the axis of the micrometer head isperpendicular to the front surface 440 of vacuum plate 395, and rigidlyconnected to surface 295 of print table 150. Non-rotating spindle 535 ofmicrometer head 520, outfitted with ball-bearing tip 536 (analogous totip 505 of micrometer head 470) is extended to bear against the frontsurface 440 of vacuum plate 395, such that, by further extension of thespindle 535, the distance 537 (between the front surface 440 of vacuumplate 395 and the straight bracket 525) is increased. Conversely,distance 537 may be decreased by retraction of spindle 535, providedonly that a manual force is applied to vacuum plate 395, in the negativey direction, to keep the spindle 535 and surface 440 in intimatecontact.

[0155] In the prototype of the invention, the micrometer heads used inmicrometer assemblies 425, 430, and 435 are sold under the tradename“Model 262ML Micrometer Head”, and are commercially available from TheL. S. Starrett Company, Athol, Mass. The ball-bearing tips 505 and 536are sold under the tradename “Model 247MA ball attachment” and arelikewise available from L. S. Starrett.

[0156] In summary, referring to FIGS. 13 and 14, the three micrometerassemblies 425, 430, and 435, because they allow adjustment of the threedistances 510, 515, and 537, provide complete, three-degree-of-freedom(xyθ) adjustment of the location and orientation of vacuum plate 395with respect to the print table 150. Thus, the xy location and θorientation of the stamp 5 can be adjusted with respect to the receiver25, because, as previously shown in FIG. 11, the stamp's back-plane 40is affixed to the vacuum plate 395, whereas, as shown in FIG. 9, thereceiver 25 is affixed to the print table 150.

[0157] This xyθ adjustability of the stamp 5 with respect to thereceiver 25 is achieved, however, without sacrificing stiffness of thestamp-to-receiver connection during the printing process. Duringprinting, the vacuum chuck 455 locks the vacuum plate 395 (and hence thestationary end 320 of the stamp's back-plane 40, as described in thenext paragraph) to the print-table's stepped-down surface 295, while thevacuum chuck 350 locks the receiver 25 to the print-table's elevatedreceiver surface 300. Thus during printing, the connection between thestamp 5 and the receiver 25 is extremely stiff—much stiffer than couldbe attained by alternative, xyθ-adjustable arrangements such asconventional stages. Such high stiffness contributes substantially tofaithful transfer of the stamp's raised pattern 15 to the receiver 25.

[0158]FIG. 15 shows a cross-sectional, exploded detail of the locatingscrews 400 (introduced in FIG. 10 and repeated in FIG. 14) and themating locating holes 405 (shown in FIG. 9), that are used in theprototype of this invention. Of course, there are many means by which aback-plane layer 40 may be simultaneously located and attached to avacuum plate 395, but the means shown are simple yet effective. Near itsstationary end 320, the back-plane 40 is provided with two or more holes540, separated by a precise center-to-center distance 545 (shown on FIG.14). Locating screws 400 are inserted through the stamp holes 540 andscrewed into the threaded portion 550 of locating holes 405. Theunthreaded portion 555 of locating holes 405 are precisely machined tomatch the distances between and diameters of the holes 540 in theback-plane 40. By these means, the two screws 400 serve not only tolocate the stamp's back-plane 40 with respect to the vacuum plate 395,but also to rigidly attach the one to the other.

[0159] As further shown in FIG. 15, the locating screw 400 has athreaded portion 560 that mates with the threaded portion 550 oflocating hole 405; a chamfered portion 565 that facilitates the entry ofscrew 400 into locating hole 405; a locating portion 570 that mates withthe unthreaded portion 555 of locating hole 405; and a head 575 whoseunder-surface comprises a relieved area 580 and a bearing area 585, andwhose top surface comprises a screw-driver slot 590 or otherscrew-torquing means. The locating portion 570 of locating screw 400 hasan axial length shorter than that of the hole's unthreaded portion 555,and a diameter that is larger than the clearance diameter of thelocating screw's threaded portion 560 but slightly less than thediameter of the hole's unthreaded portion 555, such that a slip fit ofthe screw's locating portion 570 into the hole's locating portion 555 isachieved. The diameter of screw's locating portion 570 is also slightlyless than the diameter of the stamp hole 540 in the back-plane 40, suchthat a slip fit of the screw's locating portion 570 through stamp hole540 is achieved. Regarding the head 575 of screw 400, the relieved area580 insures that the critical diameter of the screw's locating portion570 continues toward the top of the screw far enough to avoid anyunwanted interference with the aforementioned slip-fit clearances, andthe bearing area 585 affixes back-plane 40 to vacuum plate 395 whenlocating screw 400 is tightened.

[0160] 2.4 Upper Stamp Clamp

[0161]FIG. 16 shows an embodiment of an upper stamp clamp 130 (from FIG.3) looking in the positive x direction (see FIG. 4 for definition of thex axis). The upper stamp clamp shown comprises a cantilevered rod 600,two rod clamps 605 and 610, a base plate 615, two side plates 620 and625, a precision shaft 630, two ball-bushings assemblies 635 and 640,two T-bars 645 and 650, and two clamp strips 655 and 660.

[0162] Cantilevered rod 600, such as Model 45 available from NewportCorp. as used in the prototype, is rigidly affixed at its root end tofaceplate 310 of z-stage 185. Rod clamp 605, such as Newport's Model340C also used in the prototype, with its faceplate facing down, isclamped to rod 600, but is not attached to base plate 615. Rod clamp 610(identical to 605), its faceplate also facing down, is rigidly affixedto base plate 615. Side plates 620 and 625, which comprise coaxial holes670 and 675 respectively, are also rigidly affixed to base plate 615,such that precision shaft 630 (such as Thompson Model LRS-8-SS as usedin the prototype) may be suspended, parallel to the y direction, betweenthe coaxial holes 670 and 675, and affixed there by set screws.Ball-bushing assemblies 635 and 640 ride on shaft 630, such that eachassembly is free to rotate about the axis of shaft 630, and also toslide along the axis.

[0163] With the exception of the cantilevered rod 600 and the rod clamp605, the remaining parts of the upper stamp clamp (610, 615, 620, 625,630, 635, 640, 645, 650, 655, 660), hereafter referred to as the stampcarrier 680, may be removed from the printing machine 100 as a singleunit, simply by releasing the clamp 610 and sliding it off thecantilevered rod 600.

[0164] Referring to FIG. 17, there is shown a schematic, cross-sectionalview of the upper-stamp-clamp assembly, perpendicular to the axis ofshaft 630. (The rod 600 and rod clamps 605 and 610 are not shown in thisFigure). Ball-bushing assembly 635 referred to in FIG. 16, actuallycomprises ball bushing 685, such as a Thompson Model A-81420-SS as usedin the prototype, and housing 690, into which ball bushing 685 isaffixed using C rings, press fit, or any other suitable manner. Housing690 is provided with attachment surfaces 695 on opposite sides of itscylindrical surface. One end of the T-bars 645 and 650 is rigidlyattached to attachment surfaces 695 by means of small bolts. The otherends of the two T-bars 645 and 650 are likewise attached to similarflats in the ball-bushing housing belonging to the other ball-bushingassembly 640. Thus T-bars 645 and 650 are affixed parallel to shaft 630,on opposite sides of it, and the whole assembly (ball-bushing assemblies635 and 640 plus T-bars 645 and 650) can freely rotate about the axis ofshaft 630, like a paddle wheel with two paddles, and can also freelytranslate along the axis of shaft 630. The latter degree of freedomprevents shear in the stamp 5, in the y direction, as it is laid downupon receiver 25 during the printing process.

[0165] Referring to FIG. 18, the back-plane layer 40 of the stamp 5 isattached to the stamp carrier 680. This attachment is most convenientlydone prior to the attachment of the back-plane layer 40 to lower stampclamp 120, on some flat surface 705. For attaching the stamp to thestamp carrier, only one of the T-bars (either 645 or 650) and itsassociated clamp strips (655 or 660) need to be used. The unused T-barand clamp strip exist merely for rotational balance. As shown in FIG.18, suppose that T-bar 645 (and associated clamp strip 655) are selectedfor attaching the stamp. The stamp's back-plane 40 has been pre-punched,near its right end 320, with a plurality of holes 710 that align withthreaded holes 715 in T-bar 645, and also with clearance holes 720 inclamp strip 655. To begin the attachment, the back-plane's holes 710 aremanually aligned with the T-bar's threaded holes 715. Then the clampstrip 655 is laid on top of back-plane 40, thereby sandwiching theback-plane between T-bar 645 and clamp strip 655 (as also shown in FIG.17), and the clamp-strip's holes 720 are also aligned. Bolts 725 (shownin FIGS. 16 and 18) are then inserted manually through holes 720 and 710and tightened into threaded holes 715.

[0166] Referring back to FIG. 16, the assembled stamp and the stampcarrier 680 may now be mounted on the printing machine by sliding rodclamp 610 onto cantilevered rod 600. To successfully complete the latteroperation, surface 730 of base plate 615 must slide underneath faceplate665 of rod clamp 605, which can only occur when the angular orientationof rod clamp 610 matches that of clamp 605. This insures that the stampcarrier is always returned to the same angular orientation about theaxis of the cantilevered rod 600, because the orientation of clamp 605is set once and never changed.

[0167] 2.5 Carriage

[0168] As disclosed above with respect to FIG. 3, the carriage 125comprises three components: a mechanical frame 255, aprint-force-application system 260, and a stamp-control system 265.These three components are now described in detail.

[0169] 2.5.1 Carriage: Mechanical Frame

[0170] Referring to FIGS. 19 through 21, the mechanical frame 255 (fromFIG. 3) is shown in detail from various viewpoints. In FIG. 19, theentire assembly is cantilevered from faceplate 315 of x₂-stage 205 bymeans of two stiff rods (such as Newport Corp. Model 45 rods as used inthe prototype), including the upper cantilevered rod 735 and the lowercantilevered rod 740. Two rods rather than one are used for improvedstiffness, and also to insure that the assembly is aligned parallel tothe print table 150, as explained presently. Rod clamps 745, 750, and755 (such as Newport Corp. Model 340C as used in the prototype) are usedto grip the rods and allow the position of the entire assembly to beadjusted in the y direction.

[0171] Referring to FIG. 20, there is provided a T-shaped plate 760,shown to better advantage in FIG. 20, which binds the two cantileveredrods (735 and 740) together into one, stiff unit by virtue of attachingto all the rod clamps. In the embodiment shown, to allow for thestaggered arrangement of the rods, a spacer plate 770 is interposedbetween the descending tongue 765 of T-shaped plate 760 and rod clamp755. If the bolt that ties the descending tongue 765 to spacer plate 770is removed and the rod clamps 745 and 750 are loosened, the T-shapedplate 760 may be freely rotated about the axis of the upper cantileveredrod 735. This rotation is useful for servicing, because the carriage'sentire front-end assembly 775, described in more detail below, issuspended from front surface 780 of T-shaped plate 760, such that therotation provides access to the underside of assembly 775. Afterservicing, the T-shaped plate 760 and the assembly 775 may be easilyreturned to the original position, because the spacer plate 770 acts asa stop that limits rotation in the clockwise direction: the thickness ofspacer plate 770 has been machined such that, when the tongue 765 ofT-plate 760 rests against the spacer plate 770, the assembly 775 isaccurately aligned parallel to the print table 150.

[0172] Affixed to the front of the T-shaped plate 760 is amicrometer-driven stage 785 (such as that sold under the tradenameMelles Griot Model 07 XSC 007 as used in the prototype), the faceplateof which, 790, may be adjusted in the z direction by means of micrometer795. Affixed to the faceplate 790 of the stage 785 (and therebyadjustable by its micrometer 795) is a three-sided frame 800, shown tobetter advantage in FIG. 21, that resembles the frame of a sofa. Theframe 800 comprises a back plate 805 and two J-slotted side plates 810and 815 that are rigidly affixed to back plate 805 at right angles.

[0173] 2.5.2 Carriage: Need for Print-Force-Application System andStamp-Control System

[0174] The side plates 810 and 815 carry two systems important forprinting; namely, the print-force-application system 260 and thestamp-control system 265.

[0175] Referring to FIG. 22, there is shown an embodiment wherein theprint-force-application system 260 is embodied in a flat-iron 270, andthe stamp-control system 265 is embodied in a vacuum-bar 820, which ispart of the vacuum-bar assembly 280.

[0176] The simplest embodiment of the invention comprises neither theprint-force-application system 260 nor the stamp-control system 265. Insuch a simplified system, the upper stamp clamp 130 shown in FIG. 22would simply move in a trajectory 821 downward and to the right duringthe printing process, causing the inked stamp 5 to be draped upon thereceiver 25 in a manner such that the curvature of the stamp at or nearthe contact front is substantially constant throughout the trajectory.However, it has been found experimentally that, when the stamp'sback-plane layer 40 is 150-μm thick metal (e.g., Invar 36) and thelateral stamp dimensions are 381×381 mm, such a simplified system doesnot reliably produce uniform, intimate contact between the stamp 5 andthe receiver 25, and thus does not reliably provide uniform transfer ofthe ink from stamp to receiver. Apparently, although the stiffness ofthe stamp's back-plane layer 40 does induce some vertical reaction forcebetween stamp 5 and receiver 25 as they come into contact, for thedimensions stated, this reaction force is insufficient for reliableprinting.

[0177] Thus, to obtain intimate contact reliably between stamp 5 andreceiver 25, it is necessary to introduce the print-force applicationsystem 260 comprising the flat-iron 270 depicted in FIG. 22. Riding atopthe stamp during the printing process, the flat-iron 270 is essentiallya bar that moves horizontally in coordination with the motion of theupper stamp clamp 130, as described below in great detail. By virtue ofits weight, the flat-iron 270 “irons” the stamp 5 onto the receiver 25.The y dimension of the flat-iron is substantially equal to that of thepolymer layer 35 of stamp 5, such that the stamp is “ironed” by auniform weight across it's entire width. The preferred embodiment of theprint-force application system 260, of which the flat-iron 270 is themain constituent, is described in detail below in connection with FIGS.23 through 25.

[0178] Regarding the vacuum bar 820 in FIG. 22a, the invention willfunction without it, but printing accuracy is improved by its addition.The vacuum bar 820, having a vacuum chuck in its lower surface 825,attracts the backplane layer 40 of stamp 5 upward against this vacuumsurface 825, thereby creating, in the critical vicinity where the stamp5 and the receiver 25 meet, a segment of the stamp BC, shown in FIG.22b, having a well determined geometry that is constant both in time(throughout the printing process) and in space (across the width of thestamp). As long as the vacuum bar 820 retains its vacuum hold on theback-plane 40, the geometry of segment BC is guaranteed to remainconstant throughout the printing process because it is determined by thefixed geometrical relationship between the flat-iron 270 and the vacuumbar 820. The geometry of segment BC is also guaranteed to be uniformacross the width of the stamp (in the y direction) because the rigidityof the short segment BC prevents stamp sag across the width. By thuscontrolling stamp geometry, the stamp-control system 265, whose payloadis the vacuum bar 820, has been found to improve markedly the accurate(undistorted) placement of the stamp's raised pattern 15 onto receiversurface 30 of receiver 25. Details of the stamp-control system aredescribed below in connection with FIGS. 26-30.

[0179] 2.5.3 Carriage: Print-Force Application System

[0180] Although contact force between stamp 40 and receiver 25 may beapplied in a variety of ways (such as by a roller), in the preferredembodiment force is applied using the flat-iron 270, which isconstructed as shown in FIGS. 23a and 23 b.

[0181] Referring to FIG. 23a, the disembodied flat-iron is shownthree-dimensionally and inverted; in FIG. 23b, it is shown as anorthographically projected side view (also inverted). During printingand peeling, the bottom surface 830 of the flat-iron 270 floats over thesurface of the stamp's back-plane 40 on a cushion of compressed gas(e.g., air or clean nitrogen) emanating from one or more double-blindgrooves 832 cut along the centerline of surface 830. Thus, the weight ofthe flat-iron is supported by gaseous pressure. The compressed gas maybe supplied to the groove 832 via a series of small, equally spacedbleed holes 835 that connect the groove 832 to a plenum chamber 840, ahole drilled parallel to the long axis of the flat-iron 270. Compressedgas may be supplied to the plenum chamber 840 via an annular pin 845whose proximal end is pressed into a precisely bored hole 850 thatterminates one end of the plenum chamber 840. The distal end of theannular pin 845 is fitted with a pneumatic fitting 855 suitable forattachment of a flexible hose 860. The opposite end of the plenum isalso terminated with a precisely bored hole 865, which is preciselyco-axial with hole 850. Pressed into hole 865 is a solid pin 870 thatseals the plenum chamber 840.

[0182] Referring back to FIG. 21, the flat-iron's pins 845 and 870,described above, will preferably ride in J-shaped slots 875 cut intoside plates 810 and 815. Thus, during printing the flat-iron's motion isnot constrained in the vertical direction; rather, its lower surface 830is free to float over the surface of the stamp's back-plane 40 as thestamp is laid upon the receiver 25. The flat-iron floats on the cushionof compressed gas as described above. At the same time, by virtue ofbeing captured in the front and rear J-shaped slots 875, the flat-iron270 is forced, during printing, to move with the carriage in the +xdirection at the print speed ν. During times other than printing, theshort vertical segments 876 of J-shaped slots 875 provide a restingplace for the flat-iron's pins 845 and 870, so that the flat-iron'slapped surface 830 is protected from inadvertent contact with othersurfaces. In an alternative, more sophisticated, embodiment, theflat-iron would be moved automatically between the active (down)position and inactive (up) position by means of a computer-controllercam or other effective mechanism.

[0183] Referring to FIGS. 24, the print-force application system 260insures that the peak contact pressure between the stamp 5 and thereceiver 25 is uniform. FIG. 24a shows a typical cross section of theflat-iron 270 as it moves over the stamp's back-plane 40 at velocity ν(the print velocity). A cushion of compressed gas 880 emanates fromdouble-blind groove 832 and flows in both the positive and negative xdirections as shown. We assume simple Couette flow (and therefore,uniform pressure gradient) in each direction, as described, for example,in Boundary Layer Theory, by H. Schlichting, McGraw Hill (1968), thedisclosures of which are incorporated by reference herein in theirentirety. Thus, the pressure distribution 885 under the flat-iron issubstantially triangular in shape as shown in FIG. 24b: the gagepressure is p₀ at the slit (x=0) and decreases to zero (atmosphericpressure) at the edges of the flat-iron (i.e., at x=±W_(AB)/2). Thepressure causes the flat-iron to float a distance b above the surface ofthe stamp's back-plane 40. The integral of the gage pressure over x(i.e., the area of the triangle in FIG. 24b) equals theweight-per-unit-length of the flat-iron in the y direction, denoted σ,which is uniform because the flat-iron's height H and width W_(AB) areuniform. Thus, $\begin{matrix}{p_{0} = {\frac{2\sigma}{W_{AB}}.}} & (1)\end{matrix}$

[0184] From Couette theory, it may easily be shown that {dot over (Q)},the volumetric flow rate of gas per unit length in the y direction, isrelated to p₀ and to the gap size b by $\begin{matrix}{{\overset{.}{Q} = {\frac{b^{3}}{6\mu_{g}}\frac{p_{0}}{W_{AB}}}},} & (2)\end{matrix}$

[0185] where μ_(g) is the viscosity of the gas. Comparing Equations (1)and (2) yields $\begin{matrix}{{\overset{.}{Q} = {\frac{b^{3}}{3\mu_{g}}\frac{\sigma}{W_{AB}^{2}}}},} & (3)\end{matrix}$

[0186] which implies that the gap size b may be controlled by thevolumetric flow rate{dot over (Q)}. In the preferred embodiment, {dotover (Q)} is controlled directly by a flow-control device, such as arotameter (e.g., Omega Engineering Inc., Model FL-102, as used in theprototype). In the prototype of the invention, the flat-iron was made ofhardenable stainless steel such as 17-4 PH, the air-bearing surface 830was surface ground, the width of groove 832 was 1.5 mm, the flow rateper unit depth was {dot over (Q)}=0.93 cm²/sec, and the cross-sectionaldimensions of the flat-iron 270 were H=19 mm and W_(AB)=31 mm, such thatthe weight per unit length was σ=4.5×10⁴ dyne/cm. If the gas is air,then μ_(g)=2×10⁻⁴ dyne-s/cm², so the air-bearing gap for this set ofparameters, according to Equation (3), is b=49.2 μm.

[0187] 2.5.4 Carriage: Stamp-Control System

[0188] The rationale for the stamp-control system 265 has been statedabove in connection with FIG. 22. The stamp may be urged upward againstthe surface 825 shown in FIG. 22b by a variety of means, such asmagnetically if the stamp's back-plane 40 is ferromagnetic. In thepreferred embodiment it is held by vacuum, and thus the bar 820, whichspans the width of the stamp in the y direction, we refer to herein asthe “vacuum bar”.

[0189] Referring to FIGS. 25 through 29, an aluminum L-beam 890 orequivalent spans the distance between side plates 810 and 815 of thethree-sided frame 800. The L-beam 890 may be, attached to these sideplates by means of bolts 895 and 900, which pass through slots 905 and910 (visible in FIGS. 27 and 28 respectively) into threaded holes inside plates 810 and 815, thereby allowing coarse height and parallelismadjustment of vacuum bar 820 with respect to receiver surface 300 ofprint table 150. Finer (and more convenient) adjustment of height andparallelism is preferably provided by micrometer heads 915 and 920 (suchas Model 262 from The L. S. Starrett Co. as used in the prototype) thatare mounted in the horizontal flange of L-beam 890, near each end of theL-beam and fixed by setscrews 925 and 930 or equivalent means. As shownin FIG. 26, the non-rotating spindle 935 of the first micrometer head915 may be affixed to a first axle clamp 940 by means of clamp screw945. That is, after spindle 935 is inserted into hole 950, clamp screw945 is tightened, causing the width of slot 955—and hence the diameterof hole 950—to be reduced, thereby clamping spindle 935 firmly to thefirst axle clamp 940. Likewise, as shown in FIG. 27 (and also in FIGS.29a and 29 d), the non-rotating spindle 960 of the second micrometerhead 920 is affixed to the second axle clamp 965 by means of clamp screw970, which reduces the width of slot 975—and hence the diameter of hole980, thereby clamping spindle 960 firmly to the second axle clamp 965.

[0190] Referring to FIGS. 28, axle clamps 940 and 965, in addition tograsping the micrometer spindles 935 and 960 also serve to hold thevacuum bar 820 in such a way as to allow rotational adjustment of thevacuum bar about an axis parallel to the y direction. As shown in FIG.28a, a flanged ball-bearing 985 is pressed into the rear axle clamp 940.This ball bearing 985 faces the vacuum bar 820, and its inner racereceives an annular pin 990 that forms the rear end of the vacuum-bar'saxis. Likewise, a similar ball bearing (not shown) is pressed into thefront axle clamp, and receives a similar (although solid rather thanannular) pin 995 (not shown) that forms the front end of thevacuum-bar's axis. Thus the vacuum bar 820, suspended between the twoaxle clamps 940 and 965, rotates freely about the axis formed by pins990 and 995, thereby allowing angular adjustment of the vacuum bar 820about an axis parallel to the y direction. After such an adjustment, thevacuum bar's angular position θ_(CD) may be locked, such as bytightening nylon-tipped set screws 1000 and 1005 that are threaded intothe front and rear axle clamps, respectively. The setscrews' nylon tipsinsure that pins 990 and 995 are not scored when the setscrews aretightened. Slots 1010 (FIG. 26) and 1015 (FIGS. 28 and 29) provideaccess to the setscrews 1000 and 1005, so that they may be tightened orloosened.

[0191] Referring to FIGS. 29a, 29 c, and 30, the vacuum bar 820comprises two pieces: the main body 1020 and the wear layer 1025. Asdescribed above in connection with FIG. 22, the bottom surface 825 ofthe vacuum bar 820 (i.e., the bottom surface of the wear layer 1025) isin sliding contact with the stamp's back-plane 40 during printing,thereby holding the stamp up, against the force of gravity, by means ofvacuum suction. The wear layer 1025 will preferably be made of alow-friction material (such as polyethylene terephthalate,fluoropolymer, or the like) in order to minimize friction during theaforementioned sliding contact, thus eliminating “stick-slip” vibrationthat would compromise the accuracy of the printing process. Moreover,from a manufacturing viewpoint, the wear layer 1025 is a sacrificialmedium that protects the main body 1020 from sliding-contact wear; aftermany printing cycles, the easy-to-fabricate wear layer 1025, preferablyattached to the main body 1020 with double-stick tape, may be simply andeconomically replaced.

[0192] Referring to FIGS. 28 and 29a, a vacuum hose 1030 is threadedthrough slot 1015 and is attached to fitting 1035, thereby providingvacuum to cavity 1040 of rear-axle clamp 965, thence to the axial holeof annular pin 990, and thence to plenum 1045 of the vacuum-bar's mainbody 1020. The plenum 1045—an axial hole that runs the length of thevacuum-bar's main body 1020—is bored precisely coaxial near the twoends, such that, when the plenum is sealed by pins 990 and 995 pressedinto the precisely bored ends, these pins form an axis that is preciselystraight. As shown in FIG. 29, the vacuum-bar's main body 1020 has aflat surface 1050 along whose centerline is drilled a plurality of bleedholes 1055, each of which connects the plenum 1045 to the flat surface1050. Wear layer 1025, which has the same rectangular dimensions as flatsurface 1050, also comprises a plurality of bleed holes 1060 disposed inthe same pattern as the holes 1055 in the main body 1020, such that whensurface 1065 of the wear layer 1025 is aligned and affixed to flatsurface 1050 of the main body 1020 using double-stick tape (the tapehaving been perforated at the location of each of the bleed holes 1060)or other suitable attachment means, bleed holes 1055 align with bleedholes 1060, and the vacuum pressure is thus communicated from themain-body's plenum 1045 to the wear layer's bleed holes 1060. A shallow,double-blind vacuum groove 1070 is cut along the centerline ofwear-layer 1025's bottom surface 825. The vacuum groove 1070 is said tobe “double-blind” because it does not extend all the way to the ends ofsurface 825. Because the vacuum groove 1070 intersects the bleed holes1060, vacuum is communicated to this groove, and thus vacuum pressureacts on the whole area of the groove to lift the stamp's backplane 40against the vacuum surface 825.

[0193] In the preferred embodiment, the vacuum pressure thus supplied togroove 1070 in surface 825 is variable, so that the force that lifts thestamp's backplane 40 against surface 825 may be adjusted high enough tohold the backplane reliably, yet low enough to avoid excessive drag asthe vacuum bar's wear layer slides across the backplane 40. Toward thisend, the vacuum-bar pneumatics 285, mentioned above in connection withFIG. 3, will preferably comprise a venturi-based vacuum pump (such asthat used in the prototype, an Edco C4M10N-A), whose vacuum pressureoutput can be controlled by the venturi's supply pressure, which in turnis controlled by a conventional pressure regulator.

[0194] 2.6 Linear-Motion System: Hardware

[0195] Referring again to FIG. 3 and 4, stages 165, 185, and 205, whichconvert the rotary motion of motors 170, 190, and 210 into linear motionalong axes x₁, z, and x₂ respectively, are high quality,medium-precision stages. Suitable stages for use with the invention,such as used in the prototype, are manufactured by Parker/Daedal, withthe following model numbers:

[0196] Stage 165 (x₁ axis): 406600XR-MS-D3H3L2C4M4E1B1R1P2

[0197] Stage 185 (z axis): 406600XR-MS-D3H3L2C4M4E1B2R1P2

[0198] Stage 205 (x₂ axis): 406600XR-MS-D3H3L2C4M4E1B1R1P1.

[0199] Details about these stages are given in Parker/Daedal Manual No.100-9313-01, the disclosures of which are incorporated by referenceherein in their entirety.

[0200] Absolute accuracy of the stages is not critical in the quest forprinted accuracy, because the placement accuracy of printed features ofthe stamp's raised pattern 15, although critically dependent on theintegrity of the stamp's back-plane layer 40 as it is laid upon thereceiver 25, does not depend on the precise motion of the stages. Forexample, the Parker stages cited above have an absolute accuracy of only50 microns, yet the embodiment shown in FIG. 4 can faithfully replicateprinted features over an area of 271 by 203 mm with an absolute,three-sigma accuracy of less than 3 microns.

[0201] It is desirable to avoid vibration in the stages. For thisreason, in the preferred embodiment, motors 170, 190 and 210 arehigh-quality DC-servo motors rather than stepper motors. DC-servo motorshave been found to produce much smoother motion of the stamp and thecarriage as they execute the three-axis coordinated motion required tolay the stamp on the receiver. Motors suitable for use with theinvention are Parker Gemini Series motors, model SM233AE-NGSN, driven byParker Gemini drivers, model GV-L3E, such as are used in the prototype.Detailed instructions for operating the motors are in Parker manual88-017790-10C; instructions for operating the drivers are in Parkermanuals 88-017778-10B and 88-017791-10C, the disclosures of which areincorporated by reference herein in their entirety.

[0202] The motor controller 220 directs the motor drivers to turn themotors as required to create the coordinated, three-axis motions of theupper end of the stamp (axes x₁ and z) and the carriage (axis x₂), asrequired for printing and peeling. Such motor controllers are well knownin the art. A suitable motor controller for use with the invention isthat used in the prototype, a Parker model 6K8, which is described inParker manual 88-017547-10A, the disclosures of which are incorporatedby reference herein in their entirety. The Parker 6K8 controller may beprogrammed using the 6K Programming Language, as documented in twoParker publications “6K Programmer's Guide” (88-017137-10A) and “6000Series Software Reference” (88-012966-01), the disclosures of which areincorporated by reference herein in their entirety. For manual controlof the stacked pair of stages 165 and 185, it is convenient to use ahardware joystick 240, such as that used in the prototype, a Parkermodel JS-6000.

[0203] The information required to create the three-axis coordinatedmotions of axes x₁, z, and x₂ (i.e., 160, 180, 200) originates in thecomputer 225, which for example may be an IBM-compatible PersonalComputer, or any other suitable computer system. The requiredinformation is generated by software 235, which will be described indetail presently. Typically, the entire series of coordinates requiredto specify the x₁, z, and x₂ motions are calculated once by the software235, and then downloaded to the motor controller 220. Subsequently, themotors may be repeatedly driven directly from the controller 220 withoutfurther computation in the computer 225.

[0204] The computer hardware 230 comprises, in addition to the usualcomponents (microprocessor, memory, storage, I/O), a communications portsuch as an RS-232 port or other I/O means by which the computer 225communicates with the motor controller 220.

[0205] 2.7 Linear-Motion-System Software

[0206] The software 235, provides mathematical means to specify—forvirtually any type and size of stamp 5—the proper, three-axiscoordinated motions that must be executed along axes x₁, z, and x₂during the printing process to lay the stamp 5 progressively upon thereceiver 25 and subsequently to peel the stamp from the receiver. Thesethree-axis coordinated motions will hereinafter be called “the printingtrajectory” and “the peeling trajectory” respectively.

[0207] 2.7.1 Software: Motivation for a Mathematical Solution

[0208] Referring to FIG. 30, the printing trajectory must be arrangedsuch that the curved shapes that the stamp assumes throughout thetrajectory, such as those shown in FIGS. 6a, 6 b, and 6 c, allow thelower surface 825 of the vacuum bar 820 (from FIG. 22) to retain itsvacuum hold upon the stamp 5. That is, as the carriage 125 moves theflat-iron 270 and the vacuum bar 820 left-to-right along axis x₂ at auniform rate, the upper stamp clamp 130 must move the upper end of thestamp along axes x₁ and z smoothly downward and to the right so that, atevery instant, the curve of the stamp “kisses” the vacuum bar.

[0209] It may be appreciated intuitively that if the upper end of thestamp is moved downward too slowly, the vacuum bar, moving rightward,will eventually stretch and kink the stamp. Conversely, if the upper endof the stamp is moved-downward too quickly, it will eventually, byvirtue of the stamp's stiffness, pull way from the vacuum bar's hold.Preventing the latter is typically the more difficult requirement,because the vacuum-bar's vacuum chuck (i.e., the double-blind groove1070 in wear layer 1065 from FIG. 29) is deliberately weak so as tominimize sliding friction between the stamp 5 and the vacuum-bar's wearsurface 1025. Therefore, the printing trajectory will be specifiedrather carefully to insure that the stamp's natural shape remainstangent to the vacuum bar throughout the motion.

[0210] Retaining this vacuum attachment enhances printed accuracybecause holding the stamp's shape constant in the segment BC ensuresconstant curvature near the contact line (where the stamp and receivermeet, near or at point B in FIG. 22), and thereby provide uniformtensile strain in the stamp's raised pattern 15. If the printingtrajectory is incorrectly specified, the stamp will assume incorrectshapes during the printing process, eventually causing the stamp tobreak away suddenly from the vacuum bar's vacuum hold. If this happens,the accuracy of printing is ruined for two reasons: first, the printingprocess is violently disturbed when the vacuum suddenly breaks and thestamp suddenly changes its shape and, second, the stamp segment BC willthereafter assume unknown and variable geometry throughout the remainderof the printing process such that the curvature near the contact line,and therefore the tensile strain induced there in the stamp's raisedpattern 15, will not be constant.

[0211] Correct specification of the printing trajectory requires amathematical understanding of the bending mechanics of the stamp. Theanalysis given below is based on Euler's famous work on the problem of“the elastica”, as described in articles 262 and 263 of A Treatise onthe Mathematical Theory of Elasticity, by A. E. H. Love, ISBN:0486601749, the disclosures of which are incorporated by referenceherein in their entirety. Euler's analytical solution predicts thestatic shape assumed by a uniform thin elastic object, such as the stamp5, when forces are applied to its ends only. The problem considered hereis more general, because two additional elements are included: (1) theforce of gravity on the stamp, and (2) the mid-span forces applied tothe stamp by the flat-iron and vacuum bar. Consequently, the solutionsobtained here must be numerical rather than analytical, but suchnumerical solutions are rapidly generated by the computer 225.

[0212] 2.7.2 Software: Formulation of the Mathematical Solution

[0213] The stamp 5 is considered, like Euler's elastica, to be a thin,elastic object of uniform cross section, as suggested by FIG. 30. Thus,each labeled point on FIG. 30 actually represents a line in the ydirection. Point Q, which is fixed in space, is the lower edge of thestamp that is attached to the vacuum plate 395. Point E is the upperedge of the stamp that is attached to the upper stamp clamp 130 along anupper-clamp line_(E) parallel to the y axis, which translates viacrossed stages 165 and 185 along axes x₁ and z, as previously described.Point P is the axis of rotation of the upper stamp clamp's shaft 630.Points A and B are the points on the stamp that at time t are nearest,respectively, to the edges of the air-bearing-supported flat-iron 270;while at the same time points C and D touch the edges of the vacuumbar's wear layer 1025. As explained above, the flat-iron 270 and thevacuum-bar assembly 280 move in unison along axis x₂. The vacuum-barangle θ_(CD) and the height of the vacuum bar above z=0, althoughadjustable, are fixed throughout a trajectory. Hereinafter, we refer tothe various segments of the stamp as follows: QA is the “flat segment”,AB is the “air-bearing segment”, BC is the “lower segment”, CD is the“vacuum-bar segment”, and DE is the “upper segment.” By referring to QAas the “flat segment”, we implicitly assume that the flat-iron 270 isheavy enough to force all of segment QA flat onto the receiver 25. Inpractice, some or all of segment AB may also be flattened onto thestamp, as discussed presently.

[0214] During the printing process, point A (together with points B, C,and D, whose geometry is fixed relative to A) moves along the x₂ axis ata constant rate by virtue of the motor-driven x₂ stage 205. Eachposition x_(A) of point A, together with coordinates (x_(P),z_(P)) ofpoint P (the upper pivot), will be referred to as a “configuration”.That is, a configuration is specified by the three numbers(x_(A),x_(P),z_(P)). If a configuration satisfies all the constraints ofthe problem—the differential equation of the stamp and boundaryconditions, then it is considered a “viable configuration”. A printingtrajectory is simply an ensemble of viable configurations havingmonotonically increasing values of x_(A). That is, the printingtrajectory is formed by “connecting the dots” between viableconfigurations—by moving stages (x₁,z,x₂) (i.e., 165, 185, 205) incoordination with each other, with point A moving continually rightwardand point P moving continually rightward and downward, so as to performpiecewise linear interpolation between a set of viable configurations.Thus, the objective of the following mathematical analysis is tospecify, for each given value of x_(A), the required location(x_(P),z_(P)) of pivot P to produce a viable configuration.

[0215] The differential equation of the stamp, as well as the variousboundary conditions, are specified in the ensuing analysis. FollowingEuler, we treat the stamp as a bending beam that transmits an internalbending moment, a shear force, and an axial (tangential) force.Referring to FIG. 31, we use curvilinear coordinates s and θ, where s isarc length along the stamp, measured rightward from point Q. There isshown in the drawing a differential element of the stamp, ofinfinitesimal length ds and unit depth normal to the plane of thedrawing. The angle θ, measured counterclockwise from horizontal, is theangle of this differential element ds. Thus, θ is a function of s; thatis, the stamp hangs in a curve. The weight per unit area of the stamp isdenoted w, and so the force w ds acts downward on the differentialelement.

[0216] Also, acting internally on the stamp's cross-sectional face ateach end of the differential element, are the forces-per-unit-depthF_(x) and F_(z), as well as the internal bending-moment-per-unit-depthM. These internal loads are the resultants of internal,tensile/compressive stresses tangential to the stamp and shear stressesnormal to it. To simplify the mathematics, however, the loads F_(x) andF_(z) are resolved along the x and z axes rather than tangential andnormal to the stamp.

[0217] Surface stresses acting transverse to the surface of the stampare also included in this analysis. The stress normal to the face,positive downward, is denoted p (i.e., the normal force per unit depthis p ds); the stress tangential to the surface, positive toward positives, is denoted f (i.e., the tangential force per unit depth is f ds). Forsegments BC and DE, p=f=0, because no surface stresses act in theseregions.

[0218] For the air-bearing segment AB, the tangential stress f isnegligible because the air-bearing's coefficient of friction is nearlyzero. The normal pressure distribution p_(AB)(s) acting in theair-bearing segment AB comprises both a positive (i.e., downward)pressure p_(AIR)(s) from the air bearing and a negative (i.e., upward)reaction pressure p_(R)(s) from the receiver 25. The air-pressureprofile, shown in FIG. 24b, may assumed to be $\begin{matrix}{{p_{AIR}(s)} = \{ \begin{matrix}{{\frac{2\quad p_{0}}{s_{B} - s_{A}}( {s - s_{A}} )},} & {s_{A} \leq s \leq \frac{s_{A} + s_{B}}{2}} \\{{\frac{2\quad p_{0}}{s_{B} - s_{A}}( {s_{B} - s} )},} & {\frac{s_{A} + s_{B}}{2} \leq s \leq s_{B}}\end{matrix} } & \begin{matrix}( {4a} ) \\( {4b} )\end{matrix}\end{matrix}$

[0219] where the relationship between p₀ and the weight per unit lengthσ the flat-iron 270 is given by Equation (1) above. The reactionpressure p_(R)(s) may be assumed to be equal and opposite to p_(AIR)(s)over the segment AO where, as shown in FIG. 33, the stamp 5 is assumedto be in contact with the receiver 25. Thus $\begin{matrix}{{p_{R}(s)} = \{ \begin{matrix}{{- p_{AIR}},} & {s_{A} \leq s \leq s_{0}} \\{0,} & {s_{0} \leq s \leq {s.}}\end{matrix} } & \begin{matrix}( {4c} ) \\( {4d} )\end{matrix}\end{matrix}$

[0220] The net pressure in the air-bearing segment AB is therefore

p _(AB)(s)=p _(AIR)(s)−p _(R)(s).  (4e)

[0221] Referring to FIG. 32, the normal-pressure distribution p_(CD)(s)acting in the vacuum-bar segment CD comprises both the negative (i.e.,upward) vacuum pressure, −p_(vac), acting over the area C₁D₁ of thedouble-blind groove 1070 in wear layer 1025, as well as the positive(i.e., downward) reaction pressure p_(react) acting over the lowersurface 825 (CC₁ and D₁D) of wear layer 1025. Thus, assuming thatp_(react) is a uniform load, $\begin{matrix}{{p_{CD}(s)} = \{ \begin{matrix}{- p_{vac}} & {s_{C_{1}} \leq s \leq s_{D_{1}}} \\p_{react} & {s_{C} \leq s < {s_{C_{1}\quad}{and}\quad s_{D_{1}}} < s \leq {s_{D}.}}\end{matrix} } & ( {5a} )\end{matrix}$

[0222] Assuming further that the positive and negative normal forces dueto these pressures balance each other, $\begin{matrix}{{p_{react} = {( \frac{W_{C_{1}D_{1}}}{W_{CD} - W_{C_{1}D_{1}}} )p_{vac}}},} & ( {5b} )\end{matrix}$

[0223] where, as shown in FIG. 32, W_(CD) is the width of segment CD,and W_(C) ₁ _(D) ₁ is the width of the double-blind groove 1070. Thetangential stress distribution f_(CD)(s) acting in the vacuum-barsegment CD is the frictional stress associated with the reaction forcesp_(react). Thus $\begin{matrix}{{f_{CD}(s)} = \{ \begin{matrix}0 & {s_{C_{1}} \leq s \leq s_{D_{1}}} \\{\mu \quad p_{react}} & {{s_{C} \leq s < {s_{C_{1}\quad}{and}\quad s_{D_{1}}} < s \leq s_{D}},}\end{matrix} } & ( {5c} )\end{matrix}$

[0224] where μ is the coefficient of friction between the stamp 5 andthe wear layer 1025.

[0225] Although the stamp is in motion during the printing process,mathematically we assume that the process is quasi-static; that is, thatinertia forces are negligible compared to other forces. Therefore, theequations that describe the differential stamp element in FIG. 31 at anytime t during the printing process (i.e., for any configuration) are theequations of static equilibrium:

ΣHorizontal Forces=0  (6a)

ΣVertical Forces=0  (6b)

ΣCounterclockwise moments (about left end of element ds)=0  (6c)

[0226] By inspection of FIG. 31, ignoring terms of order ds², theseequations are: $\begin{matrix}{\frac{F_{x}}{s} = {{{- p}\quad \sin \quad \theta} - {f\quad \cos \quad \theta}}} & ( {7a} ) \\{\frac{F_{z}}{s} = {w + {p\quad \cos \quad \theta} - {f\quad \sin \quad \theta}}} & ( {7b} ) \\{{\frac{M}{s} + {F_{z}\cos \quad \theta} - {F_{x\quad}\sin \quad \theta}} = 0.} & ( {7c} )\end{matrix}$

[0227] The relation between moment M and curvature $\frac{\theta}{s}$

[0228] in a beam is $\begin{matrix}{M = {E\quad I\frac{\theta}{s}}} & (8)\end{matrix}$

[0229] as derived in many standard engineering texts; for example, inSection 3-5 of The Analysis of Stress and Deformation, by George W.Housner and Thad Vreeland, Jr., Library of Congress Catalog Card Number65-22615 (1975), the disclosures of which are incorporated by referenceherein in their entirety. In Equation (8), E is the Young's modulus ofthe stamp, and I is the area moment of inertia of the stamp's crosssection per unit depth in the y direction (see FIG. 30). Although thestamp 5 may comprise two layers, the back-plane layer 40 is typically somuch stiffer than the polymer layer 35 that, to an excellentapproximation, E and I may be taken to be the modulus and the moment ofinertia of the back-plane layer 40 only.

[0230] Introducing Equation (8) into Equation (7c) yields:$\begin{matrix}{{{{EI}\frac{^{2}\theta}{s^{2}}} + {F_{z}\cos \quad \theta} - {F_{x}\sin \quad \theta}} = 0} & (9)\end{matrix}$

[0231] The solution for the shape of the stamp, θ(s), is obtained bypiecewise (i.e., segment-wise) numerical integration of Equations (7a),(7b) and (9). The numerical solution is obtained by defining the vectoru of unknowns as $\begin{matrix}{u \equiv \begin{Bmatrix}u_{1} \\u_{2} \\u_{3} \\u_{4}\end{Bmatrix} \equiv \begin{Bmatrix}\theta \\\frac{\theta}{s} \\F_{x} \\F_{z}\end{Bmatrix}} & (10)\end{matrix}$

[0232] Then Equations (7a), (7b) and (9) may be written as$\begin{matrix}{{\frac{u}{s} = {F(u)}},{where}} & \text{(11a)} \\{{F(u)} \equiv \begin{Bmatrix}u_{2} \\{{\frac{u_{3}}{EI}\sin \quad u_{1}} - {\frac{u_{4}}{EI}\cos \quad u_{1}}} \\{{{- p}\quad \sin \quad u_{1}} - {f\quad \cos \quad u_{1}}} \\{w + {p\quad \cos \quad u_{1}} - {f\quad \sin \quad u_{1}}}\end{Bmatrix}} & \text{(11b)}\end{matrix}$

[0233] Given the stamp's physical properties E, I, and w, and thesurface-loading conditions p and f for each segment, Equations (11)describe how θ (as well as F_(x) and F_(z)) vary as functions ofarc-length s throughout the various segments. The surface-loadingconditions p and f in the various segments were discussed previously inconnection with Equations (4) and (5). To solve for the shape of thestamp in a given configuration, therefore, it remains only to specifythe initial values of the variables u₁ at some starting point s₀, andthen to integrate numerically the differential equations (11a) startingat s=s₀. Numerical integration of these nonlinear differential equationsmay be carried out using various methods, such as fourth-orderRunge-Kutta integration. Such methods are described, for example, inNumerical Recipes in C, by William H. Press et al., the disclosures ofwhich are incorporated by reference herein in their entirety. When theright end of a segment is reached, the values of the u₁ obtained at thefinal point (e.g., point B in segment AB) are used as the initialconditions for integration of the next segment (e.g., segment BC).

[0234] The arc-length coordinates of points A, B, C, D, and E, denoteds_(A), s_(B), s_(C), s_(D), and s_(E) respectively, must be known apriori for straightforward integration of the differential equations(11a). Of course, because segment QA is flat,

s _(A) =x _(A).  (12a)

[0235] Although the exact arc-lengths of segments AB, BC, and CD are notprecisely known a priori, these segments are typically short and thestamp's curvature therein is low. Therefore, to an excellentapproximation,

s _(B) ≈x _(A) +W _(AB);  (12b)

[0236] that is, the arc length s_(B)−s_(A) (i.e. s_(B)−x_(A)) of segmentAB is assumed to be equal to the width W_(AB) of the flat-iron itself(see FIG. 30). The arc length of segment BC, s_(C)−s_(B), isapproximately that of the straight line connecting points B and C, thus

s _(C) ≈s _(B)+{square root}{square root over ((x _(C) −x _(B))²+(z _(C)−z _(B)) ²)}.  (12c)

[0237] where the differences under the radical in Equation (12) areknown by direct measurement of the apparatus. The arc length of segmentCD is assumed to be equal to the width W_(CD) of the vacuum bar, thus

s _(D) ≈s _(C) +W _(CD).  (12d)

[0238] Finally,

s _(E) ≈L,  (12e)

[0239] where L is the total length L of the stamp, from point Q to pointE which is known by direct measurement of the stamp.

[0240] As shown in FIG. 32, point O, with arc-length coordinate s₀, isthe point where the stamp first departs from the flat condition θ=0, andthus is the starting point for integration of the differential equations(11a). Unfortunately, the location of point O is unknown a priori.However, as suggested by FIG. 32, it is typically in the segment AB. Forintegration of the differential equations, the initial conditions atpoint O (i.e., the values of the u₁ at s=s₀) are: $\begin{matrix}{{u_{0} \equiv \begin{Bmatrix}u_{10} \\u_{20} \\u_{30} \\u_{40}\end{Bmatrix} \equiv \begin{Bmatrix}\theta_{0} \\ \frac{\theta}{s} |_{0} \\F_{x_{0}} \\F_{z_{0}}\end{Bmatrix}} = \begin{Bmatrix}0 \\0 \\F_{x_{0}} \\F_{z_{0}}\end{Bmatrix}} & (13)\end{matrix}$

[0241] On the right side of Equation (13), the first two components arezero because, by hypothesis, segment AO is completely flat.

[0242] Thus there are three unknown parameters, s₀, F_(x0), and F_(z0),that must be determined for each integration of the differentialequations (11a) (i.e., for each configuration). To operate the printingmachine 100, it is necessary, for each configuration, to determinevalues of these three parameters that satisfy the following threeconditions:

[0243] 1. Stamp passes through point C. To guarantee that the stampremains attached to the vacuum bar throughout the trajectory, each stampconfiguration should pass through the point C (i.e., the trailing edgeof the vacuum bar's wear layer 1025). As shown in FIG. 34, thecoordinates (x_(C),z_(C)) of point C are given in terms of thecoordinates (x_(V),z_(V)) of the vacuum-bar's center of rotation and thehardware-adjustable rotation angle θ_(CD) by $\begin{matrix}{x_{C} = {x_{A} + x_{VA} + {H_{CD}\sin \quad \theta_{CD}} - {\frac{W_{CD}}{2}\cos \quad \theta_{CD}}}} & \text{(14a)}\end{matrix}$

$\begin{matrix}{z_{C} = {z_{V} - {H_{CD}\cos \quad \theta_{CD}} - {\frac{W_{CD}}{2}\sin \quad \theta_{CD}}}} & \text{(14b)}\end{matrix}$

[0244] where, in Equation (14a), x_(V) has been replaced by itsequivalent x_(A)+x_(VA), where x_(VA) is the constant distance shown inFIG. 30.

[0245] To determine whether a candidate stamp shape θ(s) passes throughpoint C, the stamp's curvilinear coordinates (θ,s) must be converted toCartesian coordinates (x,z) using the relations dx=cos θds and dz=sinθds. First, s_(C) is determined either approximately using Equation(12c) or more accurately as the value for which

x _(C)=∫₀ ^(s) ^(_(C)) cos θ(s)ds.  (15a)

[0246] Once s_(C) is determined, the curve θ(s) is deemed to passthrough point C if, to some tolerance,

z _(C)=∫₀ ^(s) ^(_(C)) sin θ(s)ds.  (15b)

[0247] The integrations in Equations (15a) and (15b) must be donenumerically using, for example, the well-known trapezoidal rule, whichis described in many standard texts such as Numerical Recipes in C,previously cited.

[0248] 2. Stamp angle at point C equals θ_(CD). To guarantee that thestamp remains attached to the vacuum bar, the stamp's angle at point C,θ_(C), should equal the preset value θ_(CD) discussed above inconnection with Equations (14). That is,

θ(s _(C))≡θ_(C)=θ_(CD)  (16)

[0249] where s_(C) is determined either by Equation (14c) or moreaccurately by Equation (15a).

[0250] 3. Stamp rotates freely at point E. The upper stamp clamp 130,discussed previously in connection with FIG. 17, imposes a free-rotationcondition at point E. The appropriate mathematical condition is derivedfrom FIG. 35. As illustrated in FIG. 35a, the ball-bushing assemblies635 and 640 (e.g., ball bushing 685 and ball-bushing housing 690), aswell as T-bars 645 and 650, rotate freely about axis P of shaft 630,like a paddle wheel with two paddles. Because the stamp 5 exertsforces-per-unit-y-length F_(xE), F_(zE), andbending-moment-per-unit-y-length M_(E) upon this assembly, the conditionfor free rotation demands that the sum of the moments exerted by theseloads about axis P must be zero. That is, by inspection of FIG. 35b,which is a mathematical representation of the situation:

M _(E) +F _(xE) R _(s) sin θ_(E) −F _(zE) R _(s) cos θ_(E)=0.  (17a)

[0251] Substituting Equation (8) for bending-moment-per-unit-lengthM_(E) yields $\begin{matrix}{ {{EI}\frac{\theta}{s}} \middle| {}_{E}{{{+ F_{xE}}R_{s}\sin \quad \theta_{E}} - {F_{zE}R_{s}\cos \quad \theta_{E}}}  = 0} & \text{(17b)}\end{matrix}$

[0252] which is a relationship between the stamp's angle and itscurvature at point E.

[0253] To summarize, the three conditions (15b), (16), and (17b) areused to determine the three unknown parameters s₀, F_(x0), and F_(z0).This determination can only be made by numerical iteration. To describethis process, let β be the vector of the three unknown parameters,$\begin{matrix}{{\beta = \begin{Bmatrix}s_{0} \\F_{x0} \\F_{z0}\end{Bmatrix}},} & (18)\end{matrix}$

[0254] and let the three conditions (15b), (16), and (17) be expressedas the vector condition

T=0,  (19)

[0255] where T is defined as $\begin{matrix}{{T(\beta)} \equiv {\begin{Bmatrix}{z_{C} - {\int_{0}^{s_{C}}{\sin \quad \theta \quad {s}}}} \\{\theta_{C} - \theta_{CD}} \\ {{EI}\frac{\theta}{s}} \middle| {}_{E}{{{+ F_{xE}}R_{s}\sin \quad \theta_{E}} - {F_{zE}R_{s}\cos \quad \theta_{E}}} \end{Bmatrix}.}} & \text{(20)}\end{matrix}$

[0256] As indicated in Equation (20), T is a function of β because thesolution θ(s) to the differential Equations (11a), and hence θ_(C),θ_(E), and $ \frac{\theta}{s} |_{E},$

[0257] depend on the components of β.

[0258] To determine the correct value of β—that is, the values of s₀,F_(x0), and F_(z0) that will yield a viable configuration—an initialguess, β⁽⁰⁾, is made, the differential Equations (11a) are integrated,and the resulting value of T in Equation (20) is evaluated. In general,the initial guess β⁽⁰⁾ will not produce the desired result T=0, but thecorrect value of β may be found from the initial guess (provided theinitial guess is close enough to the correct value) by means of thewell-known root-finding method known as Newton-Raphson. This entirescenario—seeking a solution satisfying given conditions (that can not beevaluated a priori) by guessing initial parameters, integrating,evaluating the conditions, and then iterating using the Newton-Raphsonmethod—is a technique called the “shooting method”, which is well knownin the art of numerical solutions for differential equations. Theshooting method, as well as the Newton-Raphson method which it employs,are described more fully, for example, in Numerical Recipes in C,previously cited.

[0259] In terms of the notation introduced above, the Newton-Raphsoniteration step that produces the new approximation for β, β^((n+1)),from the existing one β^((n)), is $\begin{matrix}{{\beta^{({n + 1})} = {\beta^{(n)} - {\lbrack \frac{\partial T}{\partial\beta} \rbrack_{\beta = \beta^{(n)}}^{- 1}{T( \beta^{(n)} )}}}},} & \text{(21a)}\end{matrix}$

[0260] and the iteration is assumed to have converged when twoconsecutive iterates are sufficiently close to each other; that is, when$\begin{matrix}{\frac{{\beta^{({n + 1})} - \beta^{(n)}}}{\beta^{(n)}} < ɛ_{C}} & \text{(21b)}\end{matrix}$

[0261] for some small number ε_(C). In Equation (21a),$\lbrack \frac{\partial T}{\partial\beta} \rbrack_{\beta = \beta^{(n)}}^{- 1}$

[0262] is the inverse of the Jacobian matrix of partial derivatives$\frac{\partial T_{i}}{\partial\beta_{j}}$

[0263] evaluated at β=β^((n)). These partial derivatives must beapproximated numerically by finite differences: $\begin{matrix}{{\frac{\partial T_{i}}{\partial\beta_{j}} = \frac{{T_{i}( {\beta_{j} + ɛ} )} - {T_{i}( \beta_{j} )}}{ɛ_{D}}},\quad i,{j = {1\text{,}2\text{,}3}}} & (22)\end{matrix}$

[0264] where ε_(D) is some small number. The above procedure—applyingthe shooting method to find β that satisfies the conditionsT=0—specifies a “viable stamp configuration” in the sense describedpreviously in connection with FIG. 30. Thus, the correct shape of thestamp, θ(s), is determined for one particular position of the carriage125; that is, for one particular value of s_(A)=x_(A). As stated above,the objective of the mathematical analysis above is to specify, for eachgiven value of x_(A), the required location (x_(P), z_(P)) of pivot P toproduce a viable configuration. The sought-after Cartesian coordinates(x_(P), z_(P)) may be determined from the stamp shape θ(s), using dx=cosθds and dz=sin θds, as follows:

x _(E)=∫₀ ^(L) cos θ(s)ds  (23a)

z _(E)=∫₀ ^(L) sin θ(s)ds  (23b)

x _(P) =x _(E) +R _(s) cos θ_(E)  (23c)

z _(P) =z _(E) +R _(s) sin θ_(E),  (23d)

[0265] where L is the total length of the stamp (from point Q to pointE), and where Equations (23c) and (23d) make use of the geometry drawnin FIG. 35b. The integrations in Equations (23a) and (23b) are performedusing the well-known trapezoidal rule, previously cited in connectionwith Equations (15).

[0266] To generate a printing trajectory, the above mathematicalsolution must be developed numerous times, once for each configuration,where a configuration is characterized by its value of x_(A) (theposition of the flat-iron 270). The array of configurations comprising aprinting trajectory has the property that x_(A) increases monotonicallythroughout the array. Let this array of x_(A) values be denoted {x_(A1),x_(A2), x_(A3), . . . , x_(AN)}. Once the first viable configuration hasbeen found, for the first value x_(A1), the associated value of β—uponwhich the Newton-Raphson method converged—is used as the initial guessfor the next configuration x_(A2). As long as x_(A2)−x_(A1) is not toolarge, this strategy guarantees that the Newton-Raphson method willeasily converge for the x_(A2) configuration. The same strategy iscontinued throughout the array: the value of β found for configurationx_(Ai) is used as the initial guess for configuration x_(A,i+1), wherei=1, . . . N−1.

[0267] 2.7.3 Software: Simplification of the Mathematical Solution

[0268] Experience with the prototype of the invention has shown that thegoal of the mathematical analysis above—providing a printing trajectoryin which the stamp 5 remains attached to the vacuum-bar's wear layer1025 throughout the trajectory—can be achieved by a simpler embodimentof the mathematical solution above. Three simplifying assumptions aremade:

[0269] 1. Known location of point O. First, assume that the contactpoint O is coincident with B (i.e., all of the air-bearing segment AB isflat). Then

s ₀ =s _(B).  (24)

[0270] Integration of the differential equations can thus proceed from aknown starting point, point B.

[0271] 2. Known conditions at point B. Further assume that the verticalcomponent of stamp force at B is zero, and that the curvature of thestamp at B may be specified a priori to be a known value κ_(B). That is,

F _(zB)=0  (25)

[0272] and $\begin{matrix}{ \frac{\theta}{s} |_{B} = {\kappa_{B}( {{\kappa_{B}\quad {specified}},{{hence}\quad {known}}} )}} & (26)\end{matrix}$

[0273] 3. Stamp segment CD lies flat on vacuum bar. Finally, assume thatover the vacuum-bar segment CD, the stamp is not curved, but rather liesflat on the surface of the vacuum-bar's wear layer 1025; that is,

θ(s)=θ_(CD)=constant, s _(C) ≦s≦s _(D).  (27)

[0274] The first two simplifying assumptions above, embodied inEquations (24) through (26), are the most essential ones. They implythat the initial conditions for integrating the differential equations(11b) are, in place of Equation (13), $\begin{matrix}{{u_{B} \equiv \begin{Bmatrix}u_{1B} \\u_{2B} \\u_{3B} \\u_{4B}\end{Bmatrix} \equiv \begin{Bmatrix}\theta_{B} \\ \frac{\theta}{s} |_{B} \\F_{xB} \\F_{zB}\end{Bmatrix}} = {\begin{Bmatrix}0 \\\kappa_{B} \\F_{xB} \\0\end{Bmatrix}.}} & (28)\end{matrix}$

[0275] That is, because κ_(B) is known, there is now only one unknowninitial parameter, F_(xB), instead of three as there were in Equation(13). Thus we may impose only one of the three conditions (15b), (16),and (17). We choose to impose (17b), a natural boundary condition thatis essential. The rest of the analysis proceeds as described previously,excepting that the three-component vectors β and T, previously definedby Equations (18) and (20), here reduce to scalars β and T:

β≡F _(xB)  (29a)

[0276] $\begin{matrix}{{ {{T(\beta)} \equiv {E\quad I\frac{\theta}{s}}} \middle| {}_{E}{{{+ F_{xE}}R_{S}\sin \quad \theta_{E}} - {F_{zE}R_{S}\cos \quad \theta_{E}}}  = 0},} & ( {29b} )\end{matrix}$

[0277] and the Newton-Raphson iteration (21) reduces to a simple, scalarNewton iteration: $\begin{matrix}{\beta^{({n + 1})} = {\beta^{(n)} - {( \frac{T}{\beta} )_{\beta = \beta^{(n)}}^{- 1}{{T( \beta^{(n)} )}.}}}} & (30)\end{matrix}$

[0278] Thus the advantage of the first two simplifying assumptionsabove, embodied in Equations (24) through (26), is that the guessworkinvolved in initiating the Newton iteration, as well as the subsequentcalculation, is minimized. Nevertheless, as is typical with finding theroots of nonlinear equations, if the initial guess for F_(xB) is too faraway from the solution, the Newton iteration will not converge, orworse, will converge to a bizarre stamp shape θ(s) that is physicallyimpossible. Examples of such bizarre shapes are shown in FIG. 36.Blindly commanding the servomotors 170, 190, and 210 to execute atrajectory composed of such bizarre stamp shapes is extremely unwise.Therefore, in the preferred embodiment, the various stamp shapes θ(s) ina trajectory are first plotted graphically, thereby simulating thetrajectory visually before executing it in hardware.

[0279] The third simplifying assumption above, embodied in Equation(27), allows the differential equations (11a) to be integratedanalytically in the vacuum bar segment CD. The scalar form of theseequations is Equations (7). Integrating the latter:

F _(xD) −F _(xC)=−∫_(s) _(C) ^(s) ^(_(D)) p _(CD)(s)sin θ(s)ds−∫ _(s)_(C) ^(s) ^(_(D)) f _(CD)(s)cos θ(s)ds  (31a)

F _(zD) −F _(zC) =w(s _(D) −s _(C))+∫_(s) _(C) ^(s) ^(_(D)) p _(CD()s)cos θ(s)ds−∫ _(s) _(C) ^(s) ^(_(D)) f _(CD)(s)sin θ(s)ds  (31b)

M _(D) −M _(C)=−∫_(s) _(C) ^(s) ^(_(D)) F _(z)(s)cos θ(s)ds+∫ _(s) _(C)^(s) ^(_(D)) F _(x)(s)sin θ(s)ds  (31c)

[0280] By virtue of assumption (27), the stamp angle θ between points Cand D is a constant, θ_(CD), and s_(D)−s_(C)=W_(CD). Thus Equations (31)become

F _(xD) −F _(xC)=−sin θ_(CD)∫_(s) _(C) ^(s) _(D) p _(CD)(s)ds−cosθ_(CD)∫_(s) _(C) ^(_(D)) f _(CD)(s)ds  (32a)

F _(zD) −F _(zC) =wW _(CD)+cos θ_(CD)∫_(s) _(C) ^(s) ^(_(D)) p_(CD)(s)ds−sin θ_(CD)∫_(s) _(C) ^(s) ^(_(D)) f _(CD)(s)ds  (32b)

M _(D) −M _(C)=−cos θ_(CD)∫_(s) _(C) ^(s) ^(_(D)) F _(z)(s)ds+sinθ_(CD)∫_(s) _(C) ^(s) ^(_(D)) F _(x)(s)ds  (32c)

[0281] In connection with Equations (5), the positive and negativenormal forces due to pressure p_(CD)(s) were assumed to balance eachother, implying

∫_(s) _(C) ^(s) ^(_(D)) p _(CD)(s)ds=0.  (33a)

[0282] Further, with the help of Equations (5b) and (5c),

∫_(s) _(C) ^(s) ^(_(D)) f _(CD)(s)ds=μp _(react)(W _(CD) −W _(C) ₁ _(D)₁ )=μW _(C) ₁ _(D) ₁ p _(vac).  (33b)

[0283] Thus, Equations (32a) and (32b) reduce to

F _(xD) −F _(xC) =−μW _(C) ₁ _(D) ₁ p _(vac) cos θ_(CD)  (34a)

F _(zD) −F _(zC) =−μW _(C) ₁ _(D) ₁ p _(vac) sin θ_(CD) +wW _(CD)  (34b)

[0284] Moreover, because segment CD is fairly short, the integrals in(32c) may be approximated by the trapezoidal rule, using only the valuesof F_(x)(s) and F_(z)(s) at the endpoints C and D. Using themoment-curvature relation (8), Equation (32c) reduces to $\begin{matrix}{{ \frac{\theta}{s} \middle| {}_{D}{- \frac{\theta}{s}} |_{C} = {\frac{W_{CD}}{2E\quad I}\{ {{( {F_{xD} + F_{xC}} )\sin \quad \theta_{CD}} - {( {F_{zD} + F_{zC}} )\cos \quad \theta_{CD}}} \}}},} & ( {34c} )\end{matrix}$

[0285] which is a statement of how the stamp's curvature at D relates tothat at C. We also recall Equation (27), which implies

θ_(D)=θ_(C)≡θ_(CD).  (34d)

[0286] By virtue of Equations (34), all four components of u, defined inEquation (10) as θ, dθ/ds, F_(x), and F_(z), are known at point D assoon as they are known at C. Therefore, no integration of thedifferential equations (11a) is required in segment CD; rather, thedifference conditions (34) are applied.

[0287] An additional advantage of the simplifications above is that onlytwo stamp segments, BC and DE, remain over which the differentialequations (11a) must be numerically integrated. Because p=f=0 for boththese segments, the third and fourth components of Equations (11a) maybe written $\begin{matrix}{\frac{u_{3}}{s} = {\frac{F_{x}}{s} = 0}} & ( {35a} ) \\{\frac{u_{4}}{s} = {\frac{F_{z}}{s} = w}} & ( {35b} )\end{matrix}$

[0288] whence

u ₃ =F _(x)=constant=F _(xB)  (36a)

u ₄ =F _(z) =w(s−s _(B))  (36b)

[0289] where Equation (36b) uses the simplifying assumption (25). ThusEquations (10) and (11b) may be replaced by the simpler equations$\begin{matrix}\begin{matrix}{{u \equiv \begin{Bmatrix}u_{1} \\u_{2}\end{Bmatrix}} = \begin{Bmatrix}\theta \\\frac{\theta}{s}\end{Bmatrix}} \\{and}\end{matrix} & ( {37a} ) \\{{F(u)} \equiv {\begin{Bmatrix}u_{2} \\{{\frac{F_{xB}}{EI}\sin \quad u_{1}} - {\frac{w( {s - s_{B}} )}{EI}\cos \quad u_{1}}}\end{Bmatrix}.}} & ( {37\text{b}} )\end{matrix}$

[0290] That is, the dimensionality of the system of differentialequations (11a) has been reduced from four to two, thereby eliminatingunnecessary calculations.

[0291] Of course, the simplifications (24) through (27) above areapproximations, and reality differs from the assumptions. First, for aprototype of the invention, the stamp under segment AB is seen toviolate slightly the simplifying assumption, stated in Equation (24),that segment AB is entirely flat. That is, in reality the stamp'sstiffness often raises point B slightly off the surface of the receiver25, and the contact front is actually somewhere in the segment AB, asassumed by the unsimplified, full theory (see FIG. 33). Moreover,because the simplified theory does not impose the conditions (15b) and(16), there is no guarantee that the various stamp configurationsthroughout the trajectory will accommodate the fixed geometricalrelationship between points B, C, and D. In other words, there is noguarantee that the stamp will remain attached to the vacuum bar, becausethe stamp's stiffness may break the vacuum seal if the stamp's shapeθ(s) is too far wrong. However, because the simplified solution doesinsist, via Equation (26), that the theoretical curvature of the stampat B, κ_(B), remains constant throughout the trajectory, the shape ofthe stamp near B tends to remain constant from one configuration toanother. Experience has shown that, because C is relatively near B forthe prototype system, the simplified solution works—the stamp doesremain attached to the vacuum bar throughout the printing trajectory,even though only a partial vacuum (e.g., p_(vac)=0.2 to 0.4 atmospheres)is used, and even though the vacuum bar is explicitly adjusted to “kiss”the stamp's shape only at the beginning of the trajectory.

[0292] The “kissing” adjustment proceeds as follows. After a printingtrajectory is calculated for the first time using the simplifiedsolution given in Equations (24)-(26), the stamp 5 and the carriage 125are moved to their calculated starting positions with the vacuum-bar'smicrometer heads 915 and 920 initially adjusted all the way up, toinsure that the vacuum bar does not initially hit the stamp. Then theheight z_(C) and the angle θ_(CD) of the vacuum bar are adjusted sothat, at the starting position, the vacuum bar is tangent to the naturalcurve of the stamp. The height z_(C) is adjusted via the micrometerheads 915 and 920; in the prototype, θ_(CD) is adjusted by loosening thenylon-tipped set screws 1000 and 1005, rotating the vacuum bar manuallyuntil its lower surface is tangent to the stamp, and then re-tighteningthe set screws. Once this is done, these adjustments never have to bealtered unless a new trajectory is computed—whenever the vacuum-bar'svacuum suction is turned on at the beginning of the print cycle, thesuction naturally “grabs” the stamp as desired. Thus printing andpeeling can be done time after time, as they would be in a manufacturingenvironment.

[0293] A typical set of parameters used in the prototype was as follows:

E=147×10⁹ N/m²

I=2.8125×10⁻¹³ m³

w=19.55 N/M²

κ_(B)=8.333 m⁻¹

s _(C) −s _(B)=0.048 m

s _(E) −s _(B)=0.605 m (at start of print process)

W _(CD)=0.016 m  (38a)

W _(C) ₁ _(D) ₁ =0.00159 m

μ=0.10

p _(vac)=20,000 N/m²

R _(s)=0.037 m

ε_(C)=0.001

[0294] ε_(D)=0.0001  (38a)

[0295] For this set of parameters, the associated solution F_(xB) uponwhich the Newton iteration (30) converges is

F_(xB)=1.67 N/m.  (38b)

[0296] Thus at the beginning of the trajectory, the stamp near thecontact line B is under slight tension (i.e., F_(xB) is positive).Further computation shows that this tension at B gradually reduces asthe printing process proceeds (i.e., as B moves rightward), becoming,for example, F_(xB)=1.27 N/m when 57 percent the stamp is flattened.These levels of tension have negligible effect on the raised pattern 15of the stamp's polymer layer 35. For example, the thickness h of thestamp implied by the conditions given in Equation (38a) is h={cuberoot}{square root over (12I)}=150×10⁻⁶ m, so the associated tensilestrain ε for the level of tension in Equation (38b) isε=F_(xB)/hE=0.076×10⁻⁶ (i.e., 76 parts per billion), which is trulynegligible.

[0297] The computed stamp shape associated with the solution associatedwith Equations (38) is plotted in FIG. 36a. For Newton's method toconverge on this desired (i.e., physically realizable) answer, theinitial guess for F_(xB), denoted (F_(xB))_(guess) must be in the range

0.0162≦(F _(xB))_(guess)≦0.0171  (39)

[0298] Otherwise, undesired solutions will be converged upon instead.For example, FIGS. 37b and 37 c show the undesired solutions convergedupon when (F_(xB) )_(guess)=0.0161 and (F_(xB))_(guess)=0.0172respectively, these values of (F_(xB) )_(guess) being just outside thenarrow range given in Equation (39b). Thus, finding a desirable solutionfor F_(xB) the first time can be tedious—which is why Equations (38) and(39) are given herein. However, once a desirable solution is found forsome set of parameters, other solutions (for other sets of parameters)may be found by perturbation, wherein one of the parameters isperturbed, and(F_(xB))_(guess) is taken to be the known solution F_(xB)for the unperturbed set of parameters. This very process is used indeveloping a print trajectory, wherein the only parameter that changesfrom configuration to configuration is the stamp length s_(E−s) _(B)ahead of the flat-iron 270.

[0299] 3. Results: Printed Accuracy

[0300] One of the objectives of the printing machine 100 is to obtainthe greatest possible printing accuracy. Results shown below demonstratethe level of accuracy that has been achieved with a prototype of thisinvention. For reasons discussed below, it is anticipated that evengreater accuracy is possible by using superior raw materials for thestamp's back-plane layer 40.

[0301] Perfect accuracy would be attained if the location of eachprinted feature on the receiver 25 were identical to that of thecorresponding feature on a reference substrate. This reference substratemay be the stamp 5; alternatively, it may be the “master” from which thestamp is made. In reality, feature locations on the receiver 25 differslightly, in both the x and y directions, from those on the referencesubstrate. Experimentally, these feature-placement errors may bemeasured using a high-precision, coordinate measuring machine such as aNikon-3i, which is well known in the art of semiconductor lithography.Results of such measurements may be presented as an array of arrows,such as arrows 1100, 1105, and 1110 on FIG. 37a, that compare variousfeature locations on a printed receiver to their counterparts on thereference substrate. Such plots, well known in the art of semiconductorlithography, are described, for example, in J. D. Armitage Jr. and J. P.Kirk, Proc. SPIE, 921 (1988). The arrows' tails show the locations offeatures (such as 1115, 1120, and 1125) on the reference substrate. Ifdrawn to scale as in FIG. 37a, the arrows' heads show the location ofthe corresponding features (such as 1130, 1135, and 1140) on the printedreceiver. Thus the magnitude and direction of the arrows representfeature-placement error. In practice, however, the errors are so smallcompared to the size of the array that, in order to see the arrowsgraphically, it is necessary to exaggerate their size greatly, assuggested in FIG. 37b. In such a plot, the arrows only give thedirection of the feature-placement error; the magnitude is given bycomparison to a reference arrow 1145, which is annotated by the lengthL₁ that the reference arrow represents. In contrast, the actual scale ofthe substrate is given by the reference scale 1150, which is delimitedby two tick marks and annotated with the dimension L₂ between the ticks.

[0302] If the stamp itself is used as the reference substrate, thenmeasured errors are attributable solely to the printing process.However, if the reference substrate is the “master” from which the stampis made, then measured errors are attributable to both the stamp-makingprocess and the printing process together. In all of the results below,the latter technique was used, because the stamp was too large tomeasure in the Nikon-3i machine. Thus, stamp-making and printing errorsare inextricably combined in the results below.

[0303] The prototype of the invention has been used to make a number ofprototype prints of size 381×381 mm. In these prototypical prints, thereceiver 25 is a piece of glass sputter-coated with a thin film oftitanium/gold, the ink 20 is hexadecanethiol, and the stamp's polymerlayer 35 is polydimethylsiloxane (PDMS). This combination of materialsfor receiver, ink and polymer layer is well-known in the art, of thetype having been pioneered by Kumar and Whitesides, previously cited.The prototype stamps' polymer layer 35 is 750 μm thick; the back-planelayer 40, made of Invar 36, is 150 μm thick.

[0304] For each prototypical print, error measurements such as thosedescribed above have been made over an array of approximately 2200features regularly distributed over a 271-by-203-mm rectangle that iscentered on the print. The results of such measurements on eight suchprints, made from two different stamps designated A and B, are shown asFIGS. 39 through 46. Occasionally there are missing data points (i.e.,missing arrows) on these plots. This missing data has no significanceexcept that the Nikon-3i could not optically interpret the featuresthere. The data is summarized by statistics given in Table 1. For eachof the prototypical prints listed, the print velocityν, defined above inconnection with FIG. 24a, was 10 mm/s. However, other prints having ν=40mm/s were made and measured. At this higher print speed, the placementaccuracy for a given stamp and print direction is nearly identical tothat shown in Table 1. TABLE 1 Measured Accuracy of Printing with aPrototype of the Invention Std. Deviations of Placement Print VacuumFIG. Receiver Print Error (μm) Stamp Table Bar? No. No. Direction σ_(x)σy A Al No 38 1 Forward 2.58 2.07 Ground 39 2 Forward 2.46 1.36 Al 40 3Forward 1.61 0.65 Granite Yes 41 4 Forward 1.70 0.69 42 5 Reverse 1.420.80 B Granite Yes 43 6 Reverse 0.99 0.93 44 7 Reverse 1.08 0.97 45 8Forward 1.78 2.08

[0305] Column 2 of Table 1 specifies the type of print table used,thereby giving a qualitative assessment of its flatness, the importanceof which has previously been discussed in connection with FIG. 9. Theprint table for receiver No. 1 was a piece of cast aluminum, commonlyknown as “jig plate”. The improved, flatter print table used forreceivers No. 2 and No. 3 was the same piece of aluminum after surfacegrinding. For receivers No. 4 through No. 8, a lapped granite printtable, as described above for the preferred embodiment, was used toobtain the ultimate in flatness. Column 3 of Table 1 specifies whetheror not the vacuum bar, described above in connection with FIGS. 26through 30, was used during printing.

[0306] Column 6 of Table 1 specifies the “Print Direction”, the meaningof which is explained by FIG. 43. As shown, let the two x-facing ends ofthe stamp 5 be denoted S and T respectively. Receivers 1, 2, 3, 4, and 8in Table 1 were printed with their respective stamps mounted in theprinting machine as shown in FIG. 43a—with the S end (at left in thedrawing) mounted on the lower stamp clamp's vacuum plate 395, and the Tend (at right in the drawing) mounted in the stamp carrier 680. Thisconfiguration is designated as the “forward” print direction. Receiver5, 6, and 7 in Table 1 were printed with the stamps mounted in thereverse direction, as shown in FIG. 43b—with the T and S ends reversed.This reversal was possible because the stamps were made symmetricallyleft to right in the prototype, with appropriate mounting holes in bothends, and the appropriate extra length of back-plane (only required onthe top end) on both ends. On FIGS. 39-46, the orientation of theprinted pattern is presented as invariant from drawing to drawing,regardless of print direction. However, as shown on the figures, thedirection of printing (motion of flat-iron 270 and vacuum bar 820) isleft to right (forward) on FIGS. 39, 40, 41, 42 and 46, whereas it isright to left (reverse) on FIGS. 43, 44, and 45. Of course, in realitythe motion and operation of the printing machine is always the same. Itis only the stamp that is turned around.

[0307] Columns 7 and 8 of Table 1 give standard deviations of theplacement-error measurements over the entire array of features. As iscommon in the art of such error measurements (see J. Kirk, previouslycited), the rigid-body (x,y) displacements of the array as a whole, therigid-body rotation of the array, and a uniform “magnification” errorhave all been statistically removed from the plots and from the standarddeviations reported in Table 1, because these components of error, beingeasily compensated, are of little interest in practice. Inasmuch as therigid-body components of error have been removed, the random variableswhose standard deviations are reported in Table 1 have zero mean, suchthat “three sigma error bars” are simply ±3σ.

[0308] For the best print, receiver No. 6 (FIG. 43), these ±3σ errorbars are slightly less than ±3 μm. To our knowledge, this is the mostaccurate, large-scale mechanical printing known in the art. Moreover,even greater accuracy should be attainable with improvements instamp-making, as described below.

[0309] Furthermore, the error statistics quoted in Table 1 include asignificant component of error called differential magnification (see J.Kirk, previously cited). If this component is removed for receiver No.6, for example, the statistics are reduced to σ_(x)=0.83 μm andσ_(y)=0.61 μm. For some applications, the latter statistics are morerelevant than those in Table 1. For example, when a pattern printed withthe printing machine 100 is subsequently overlaid with another layer bymeans of standard step-and-repeat optical lithography, it has been foundthat the ±3σ overlay errors closely track the smaller numbers, such asthose above, where the differential magnification error has beenremoved. The reason for this is that the optical stepper measuresmagnification anisotropy via three or more fiducials, and corrects forthe differentiation magnification error by stepping anisotropically inthe two directions. This correction works best if the stepper's shotsize is small; worse if it is large. For the prototype prints (271×203mm), the differential magnification error was found to be virtuallyeliminated from the overlay error when the optical stepper has a shotsize of 32×32 mm.

[0310] Based on experience with several hundred prints, fourconclusions, illustrated by FIGS. 39-46, may be drawn:

[0311] 1. The flatness of the print table is significant. Thisconclusion is illustrated by comparing FIGS. 39 and 40, which showerrors for receiver 1 and 2 respectively. The only difference betweenthese two cases is the flatness of the print table. As shown by theassociated statistics in Table 1, the flatter, surface-ground aluminumprint table, although it produced only a slight improvement in σ_(x),produced a significant, 34-percent improvement in σ_(y) (2.07 μm reducedto 1.36 μm). The granite table appears to have produced no furtherimprovement vis-à-vis the ground aluminum, as shown by a comparison ofFIGS. 41 and 42 (as well as the associated statistics), which are quitesimilar.

[0312] 2. The stamp-control system, embodied in the vacuum bar, isimportant. This conclusion is illustrated by comparing FIGS. 40 and 41,where the only difference is the absence or presence of the vacuum bar.As shown by the associated statistics in Table 1, the vacuum barproduced a 35-percent improvement in σ_(x) (2.46 μm reduced to 1.61 μm),and a 52-percent improvement in σ_(y) (1.36 μm reduced to 0.65 μm).

[0313] 3. With fixed conditions, printing repeatedly with the same stampin the same direction always yields nearly identical results. Thisconclusion is illustrated by comparing FIGS. 44 and 45, which were bothobtained by printing with stamp B in the “Reverse” direction. Noticethat the “fingerprint” of errors—the shape of the vector map—is verysimilar for these two plots, and the statistics differ by only a fewpercent. Such repeatability (and often better) was observed consistentlyover dozens of prints, even for prints made months apart with the samestamp.

[0314] 4. With fixed conditions, printing with different stamps, or evenwith the same stamp in opposite directions, yields significantlydifferent results. This conclusion is illustrated by the followingpaired comparisons: FIG. 41 vs. 42 (same stamp A, opposite directions);FIG. 44 vs. 45 (same stamp B, opposite directions); FIG. 41 vs. 45(different stamps, both printed in the forward direction); and FIG. 42vs. 43 (different stamps, both printed in the reverse direction). Innone of these cases do the error maps compare in any simple way. Forexample, when the print direction is reversed, the fingerprint is notsimply rotated 180 degrees, as it would be if the errors wereattributable to the printing process alone.

[0315] The most likely explanation for observations Nos. 3 and 4 aboveis that, for the prototype system, the sheet-metal material (Invar 36,150-μm thick) from which the stamp's back-plane layer 40 is made is notperfectly planar. Rather, the metal has slight bumps caused by anomaliesin its manufacture. With prototypical Invar sheets resting on a granitesurface plate, bump amplitudes up to 2.8 mm and wavelengths from 90 to200 mm have been measured. During printing, these bumps are “ironed out”by the flat-iron 270, thereby causing lateral displacement of materialpoints on the surface of the polymer layer, since the polymer followsthe much-stiffer metal.

[0316] This mechanism easily explains the observation that differentstamps have different error fingerprints, inasmuch as each stamp'sback-plane is a unique piece of sheet metal having a unique distributionof bumps that are “ironed out” in unique ways. This mechanism alsoexplains the excellent print-to-print repeatability when a given stampis repeatedly printed in the same direction—apparently, the bumps “ironout” the same way every time. Less obviously, this mechanism alsoexplains the observation that, with the same stamp, the errorfingerprint depends on print direction. The reason for this is shown inFIG. 46. Suppose there is a single bump 1155 located near the T end ofthe stamp. When printing is performed from S to T, as in FIG. 46a, thebump 1155 will not affect printed errors except near the right end ofthe receiver 25, because the bump is not encountered by the flat-iron270 until late in the printing process. However, if the same stamp isprinted in the reverse direction, from T to S, as shown in FIG. 46b,bump 1155 is flattened by flat-iron 270 nearly at the outset of theprinting process, and thus its effect can spread throughout the printedpattern on the receiver. With many bumps participating and interactingon a real stamp, it can be imagined that complex patterns of errors,such as those observed in FIGS. 42-46, may result, and may differmarkedly for the two print directions.

[0317] For the above reasons, the accuracy of printing with the currentinvention may be improved, beyond that given for receiver No. 6 in Table1, by improving the quality of the back-plane material 40. The prototypestamps were made with “off-the-shelf” material. However, as is wellknown in the art of sheet-metal manufacture, post-processing called“leveling” may be applied to reduce the bumpiness of metal sheets. Thebasic idea is to pull the material beyond its yield point in order tostretch out the extra arc length of material in the bumps. Various typesof this procedure are described, for example, in The Metals Handbook,Ninth Edition, Volume 14: Forming and Forging, ASM International, S. L.Semiatin, Joseph R. Davis, et al., editors (1988), the disclosures ofwhich are incorporated by reference herein in their entirety.

[0318] Although not mentioned in The Metals Handbook, a non-dimensionalmeasure of bumpiness used by the sheet-metal industry is $\begin{matrix}{{I \equiv {10^{5}\frac{\Delta \quad s}{\lambda}}},} & ( {40a} )\end{matrix}$

[0319] where Δs is the extra arc length in a sinusoidal bump ofamplitude A and wavelength λ. This extra arc length may easily be shownmathematically to be related to the other parameters by $\begin{matrix}{{\Delta \quad s} = {\frac{\pi^{2}}{4}{\frac{A^{2}}{\lambda}.}}} & ( {40\quad b} )\end{matrix}$

[0320] By this measure, the off-the-shelf material referred toabove—typical of that from which prototype stamps A and B were made—hasworst-case bumps having I=10 to I=50. In contrast, industrial levelingof sheet-metal material is available to reduce such worst-case bumps byan order of magnitude, to I=1. Thus, although the current inventionalready has been demonstrated to yield the most accurate large-scalemechanical printing known in the art, further improvement in printedaccuracy is available via a well-known industrial process.

[0321] Alternatively, the invention may be used with other back-planematerials, such as thin sheets of glass or ceramic, whose inherentflatness may be superior to sheet metal. Although the use of suchalternative materials may require slight changes in the above detaileddescription, these changes would not depart from the spirit and scope ofthe invention.

[0322] It is to be understood that all physical quantities disclosedherein, unless explicitly indicated otherwise, are not to be construedas exactly equal to the quantity disclosed, but rather about equal tothe quantity disclosed. Further, the mere absence of a qualifier such as“about” or the like, is not to be construed as an explicit indicationthat any such disclosed physical quantity is an exact quantity,irrespective of whether such qualifiers are used with respect to anyother physical quantities disclosed herein.

[0323] While preferred embodiments have been shown and described,various modifications and substitutions may be made thereto withoutdeparting from the spirit and scope of the invention. Accordingly, it isto be understood that the present invention has been described by way ofillustration only, and such illustrations and embodiments as have beendisclosed herein are not to be construed as limiting to the claims.

What is claimed is:
 1. A printing apparatus, comprising: a print surfacelying in a print plane defined by an imaginary x-axis and y-axis, theprint surface having an outward normal pointing in the positivedirection along an imaginary z-axis, such that the x-axis, y-axis, andz-axis are substantially orthogonal to one another; a lower stamp clampdisposed adjacent to the negative-x edge of the print surface; an upperstamp clamp, moveable in two dimensions in a trajectory plane defined bythe x-axis and z-axis; a stamp comprising a flexible material, the stamphaving a first end attached to the lower stamp clamp and a second endattached to the upper stamp clamp, such that a cross section of thestamp parallel to the trajectory plane forms an arc extending from anorigin point Q on the lower stamp clamp having (x,z) coordinates (0,0)to point E on the upper stamp clamp, this arc being described by themathematical function θ(s), where s is the curvilinear distance alongthe arc measured from point Q, and θ is the angle between the printplane and an imaginary line, the imaginary line being tangent to thecross section of the stamp at s; and wherein, during a print operation,the upper stamp clamp is moved in a trajectory comprising a plurality ofxz positions of the upper clamp stamp that blend into a substantiallycontinuous motion, the trajectory being effective in laying the stampdown smoothly and flat upon the print surface in a manner such that amoving contact front between the stamp and the print surface is created,the contact front being disposed substantially along a linecharacterized by a contact-front coordinate so xo that increases as thetrajectory progresses, the trajectory also being effective in causingthe curvature $\frac{\theta}{s}$

of the stamp at or near the contact front to be substantially constantthroughout the motion.
 2. The apparatus of claim 1 further comprising aprint-force-application system effective in pressing the stamp againstthe print surface, and defining an approximate contact front disposedsubstantially along a line_(B) parallel to the y-axis in the xy plane,the line_(B) intersectin g the trajectory plane at (x,z)=(x_(B), 0), theapproximate-contact-front x-coordinate x_(B) increasing as thetrajectory progresses and being substantially equal, at any stage of thetrajectory, to the arc-length coordinate s_(B) of point B, inasmuch asthe arc of the stamp is assumed to be substantially flat over thesegment from point Q to point B.
 3. The apparatus of claim 1 furthercomprising a stamp-control system movable along the x-axis; wherein,throughout the trajectory, each xz position of the upper stamp clamp isa function of the displacement x_(C) of the stamp-control system alongthe x-axis; the trajectory being effective in laying the stamp down uponthe print surface such that the stamp is in continuous contact with acontact surface of the stamp-control system throughout the trajectory,the location of the contact surface being characterized by an arc-lengthcoordinate s_(C) that increases as the trajectory progresses.
 4. Theapparatus of claim 3 wherein the stamp-control system is disposed alonga line_(C) parallel to the y-axis, line_(C) intersecting the trajectoryplane at point C having coordinates x_(C) and z_(C), where z_(C) is afixed, positive z-coordinate during any one printing operation, whereasx_(C) increases as the trajectory progresses, in coordination with thecontact-front coordinate x₀.
 5. The apparatus of claim 4 wherein thecontact surface of the stamp-control system is a plane delimited in thex direction by two lines _(C) and _(D) sep a rated by a fixed distanceW_(CD), these lines being parallel to the y-axis and intersecting thetrajectory plane at points C and D respectively, these points havingcoordinates (x_(C),z_(C)) and (x_(D),z_(D)) respectively, such that thecontact surface is defined by the three parameters (x_(C),z_(C),θ_(CD)),where$\theta_{CD} \equiv {\tan^{- 1}( \frac{z_{D} - z_{C}}{x_{D} - x_{C}} )}$

is the angle between the contact surface and the print plane, and suchthat the stamp angle θ(s) between arc-length coordinates s=s_(C) ands=s_(D) is substantially equal to θ_(CD); that is, θ(s)≈θ_(CD) for s_(C) ≦s≦s _(D).
 6. The apparatus of claim 1 wherein the upper stampclamp is pivoted about a pivot line_(P) parallel to the y axis andintersecting the xz plane at point P having coordinates x_(P) and z_(P);the stamp attaching to the upper stamp clamp along an upper-clampline_(E) parallel to they axis and intersecting the xz plane at point Ehaving coordinates x_(E) and z_(E); the upper-clamp line_(E) beingdisposed on the upper stamp clamp at a radius R_(s) from the pivotline_(P), such that the total arc length s_(E) from the lower stampclamp to the line_(E) is s_(E) L, where L is the known, free length ofthe stamp; and wherein the stamp attaches to the upper-clamp line_(E) atan angle θ_(E)≡θ(L).
 7. The apparatus of claim 6 wherein the trajectorycomprises a plurality of configurations, each configuration described bythe coordinate s₀ x₀ of the contact front and by correspondingcoordinates x_(P), z_(P) of the pivot line given by the equations x _(P)=x _(E) +R _(s) cos θ_(E) z _(P) =z _(E) +R _(s) sin θ_(E), wherex_(E)=∫₀ ^(L) cos θ(s)ds and z_(E=∫) ₀ ^(L) sin θ(s)ds, and where themathematical function θ(s) describing the shape of the arc for a givenconfiguration is assumed to be θ(s)=0 for 0≦s≦s ₀, whereas for s>s₀,θ(s) is determined by solution of the differential equations${\frac{u}{s} = {F(u)}},$

the lower-end boundary conditions ${{u_{0} \equiv \begin{Bmatrix}u_{10} \\u_{20}\end{Bmatrix} \equiv \begin{Bmatrix}\theta_{0} \\ \frac{\theta}{s} |_{0}\end{Bmatrix}} = \begin{Bmatrix}0 \\\kappa_{0}\end{Bmatrix}},$

and the upper-end boundary condition $\begin{matrix}{{ {{T(\beta)} \equiv {{EI}\frac{\theta}{s}}} \middle| {}_{E}{{{+ F_{X0}}R_{S}\sin \quad \theta_{E}} - {{w( {s - s_{0}} )}R_{s}\cos \quad \theta_{E}}}  = 0},} \\{{wherein}} \\{{u \equiv \begin{Bmatrix}u_{1} \\u_{2}\end{Bmatrix} \equiv \begin{Bmatrix}\theta \\\frac{\theta}{s}\end{Bmatrix}},} \\{{{F(u)} \equiv \begin{Bmatrix}u_{2} \\{{\frac{F_{X0}}{EI}\sin \quad u_{1}} - {\frac{w( {s - s_{0}} )}{EI}\cos \quad u_{1}}}\end{Bmatrix}},}\end{matrix}$

κ₀ is a specified curvature at point O, the parameter β≡F_(x0), unknowna priori, is the internal x-directed force acting on the stamp's crosssection at s=s₀ per unit depth of the stamp in the y direction, E isYoung's modulus of the stamp, I is the area moment of inertia of thestamp's cross section per unit depth in the y-direction, and w is theweight per unit area of the stamp; and wherein for each configurationthe solution for x_(P) and z_(P) is derived by means of the “shootingmethod”, whereby an initial value β⁽⁰⁾ of βis guessed, the differentialequations are solved to yield T (β⁽⁰⁾) and$\lbrack \frac{\partial T}{\partial\beta} \rbrack_{\beta = \beta^{(0)}}^{- 1},$

 Newton iteration$\beta^{({n + 1})} = {\beta^{(n)} - {\lbrack \frac{\partial T}{\partial\beta} \rbrack_{\beta = \beta^{(n)}}^{- 1}{T( \beta^{(n)} )}}}$

 is applied to obtain a refined value β⁽¹⁾ of the unknown parameter β,whereupon the differential equations are solved again; this iterationprocedure being applied repeatedly until the correct auxiliary boundarycondition T(β)=0 is achieved to within some tolerance.
 8. The apparatusof claim 2 wherein the upper stamp clamp is pivoted about a pivotline_(P) parallel to the y axis and intersecting the xz plane at point Phaving coordinates x_(P) and z_(P); the stamp attaching to the upperstamp clamp along an upper-clamp line_(E) parallel to the y axis andintersecting the xz plane at point E having coordinates x_(E) and z_(E);the upper-clamp line_(E) being disposed on the upper stamp clamp at aradius R_(s) from the pivot line_(P), such that the total arc lengths_(E) from the lower stamp clamp to the line_(E) is s_(E) L, where L isthe known, free length of the stamp; and wherein the stamp attaches tothe upper-clamp line_(E) at an angle θ_(t)≡θ(L).
 9. The apparatus ofclaim 8 wherein the trajectory comprises a plurality of configurations,each configuration described by the coordinate s_(B) x_(B) of theapproximate contact front and by corresponding coordinates x_(P), z_(P)of the pivot line given by the equations x _(P) =x _(E) +R _(s) cosθ_(E) z _(P) =z _(E) +R _(s) sin θ_(E) where x_(E=∫) ₀ ^(L) cos θ(s)dsand z_(E)=∫₀ ^(L) sin θ(s)ds,and where the mathematical function θ(s)describing the shape of the arc for a given configuration is assumed tobe θ(s)=0 for 0≦s≦s _(B), whereas for s>s_(B), θ(s) is determined bysolution of the differential equations ${\frac{u}{s} = {F(u)}},$

the lower-end boundary conditions ${{u_{B} \equiv \begin{Bmatrix}u_{1B} \\u_{2B}\end{Bmatrix} \equiv \begin{Bmatrix}\theta_{B} \\ \frac{\theta}{s} |_{B}\end{Bmatrix}} = \begin{Bmatrix}0 \\\kappa_{B}\end{Bmatrix}},$

and the upper-end boundary condition${ {{T(\beta)} \equiv {{EI}\frac{\theta}{s}}} \middle| {}_{E}{{{+ F_{XB}}R_{S}\sin \quad \theta_{E}} - {{w( {s - s_{B}} )}R_{S}\cos \quad \theta_{E}}}  = 0},{wherein}$${u \equiv \begin{Bmatrix}u_{1} \\u_{2}\end{Bmatrix} \equiv \begin{Bmatrix}\theta \\\frac{\theta}{s}\end{Bmatrix}},{{F(u)} \equiv \begin{Bmatrix}u_{2} \\{{\frac{F_{XB}}{EI}\sin \quad u_{1}} - {\frac{w( {s - s_{B}} )}{EI}\cos \quad u_{1}}}\end{Bmatrix}},$

κ_(B) is a specified curvature at point B, the parameter β≡F_(xB),unknown a priori, is the internal x-directed force acting on the stamp'scross section at S=s_(B) per unit depth of the stamp in the y direction,E is Young's modulus of the stamp, I is the area moment of inertia ofthe stamp's cross section per unit depth in the y-direction, and w isthe weight per unit area of the stamp; and wherein for eachconfiguration the solution for x_(P) and z_(P) is derived by means ofthe “shooting method”, whereby an initial value β⁽⁰⁾ of βis guessed, thedifferential equations are solved to yield T (β⁽⁰⁾) and$\lbrack \frac{\partial T}{\partial\beta} \rbrack_{\beta = \beta^{(0)}}^{- 1},$

 Newton iteration$\beta^{({n + 1})} = {\beta^{(n)} - {\lbrack \frac{\partial T}{\partial\beta} \rbrack_{\beta = \beta^{(n)}}^{- 1}{T( \beta^{(n)} )}}}$

 is applied to obtain a refined value β⁽¹⁾ of the unknown parameter β,whereupon the differential equations are solved again; this iterationprocedure being applied repeatedly until the correct auxiliary boundarycondition T(β)=0 is achieved to within some tolerance.
 10. The apparatusof claim 5 wherein the upper stamp clamp is pivoted about a pivotline_(P) parallel to the y axis and intersecting the xz plane at point Phaving coordinates x_(P) and z_(P); the stamp attaching to the upperstamp clamp along an upper-clamp line_(E) parallel to the y axis andintersecting the xz plane at point E having coordinates x_(E) and z_(E);the upper-clamp line_(E) being disposed on the upper stamp clamp at aradius R_(s) from the pivot line_(P), such that the total arc lengths_(E) from the lower stamp clamp to the line_(E) is s_(E) L, where L isthe known, free length of the stamp; and wherein the stamp attaches tothe upper-clamp line_(E) at an angle θ_(E)≡θ(L).
 11. The apparatus ofclaim 10 wherein the trajectory comprises a plurality of configurations,each configuration described by the coordinate s₀ x₀ of the contactfront and by corresponding coordinates x_(P), z_(P) of the pivot linegiven by the equations: x _(P) =x _(E) +R _(s) cos θ_(E) z _(P) =z _(E)+R _(s) sin θ_(E), where x_(E)=∫₀ ^(L) cos θ(s)ds and z_(E)=∫₀ ^(L) sinθ(s)ds, and where the mathematical function θ(s) describing the shape ofthe arc for a given configuration is assumed to be θ(s)=0 for 0≦s≦s ₀,whereas for s>s₀, θ(s) is determined by solution of the differentialequations ${\frac{u}{s} = {F(u)}},$

the lower-end boundary conditions ${{u_{0} \equiv \begin{Bmatrix}u_{10} \\u_{20} \\u_{30} \\u_{40}\end{Bmatrix} \equiv \begin{Bmatrix}\theta_{0} \\ \frac{\theta}{s} |_{0} \\F_{X_{0}} \\F_{Z_{0}}\end{Bmatrix}} = \begin{Bmatrix}0 \\0 \\F_{X_{0}} \\F_{Z_{0}}\end{Bmatrix}},$

and the auxiliary boundary conditions T(β)=0, T(β) = 0, wherein${u \equiv \begin{Bmatrix}u_{1} \\u_{2} \\u_{3} \\u_{4}\end{Bmatrix} \equiv \begin{Bmatrix}\theta \\\frac{\theta}{s} \\{F_{x}(s)} \\{F_{z}(s)}\end{Bmatrix}},{{F(u)} \equiv \begin{Bmatrix}u_{2} \\{{\frac{u_{3}}{EI}\sin \quad u_{1}} - {\frac{u_{4}}{EI}\cos \quad u_{1}}} \\{{{- {p(s)}}\sin \quad u_{1}} - {{f(s)}\cos \quad u_{1}}} \\{w + {{p(s)}\cos \quad u_{1}} - {{f(s)}\sin \quad u_{1}}}\end{Bmatrix}},{{T(\beta)} \equiv \begin{Bmatrix}{z_{C} - {\int_{0}^{s_{C}}{\sin \quad {\theta (s)}{s}}}} \\{\theta_{C} - \theta_{CD}} \\ {{EI}\frac{\theta}{s}} \middle| {}_{E}{{{+ F_{XE}}R_{S}\sin \quad \theta_{E}} - {F_{ZE}R_{S}\cos \quad \theta_{E}}} \end{Bmatrix}},$

and wherein F_(X)(s) and F_(Z)(s) are functions of s describing theinternal x-directed and z-directed forces acting on the stamp's crosssection at s per unit depth of the stamp in the y direction,F_(XE)≡F_(X)(s_(E)), F_(ZE)≡F_(Z)(s_(E)), βis a vector of parametersthat are unknown a priori, ${\beta = \begin{Bmatrix}s_{0} \\F_{X0} \\F_{Z0}\end{Bmatrix}},$

s₀ is the aforementioned arc-length coordinate of the contact front,F_(X0)≡F_(X)(s₀), F_(Z0)≡F_(Z)(s₀), E is Young's modulus of the stamp, Iis the area moment of inertia of the stamp's cross section per unitdepth in the y-direction, w is the weight per unit area of the stamp,p(s) and f(s) are functions of s describing forces applied normal to thestamp and tangential to the stamp respectively by theprint-force-application system, the stamp-control system and the printsurface, s_(C) is the value of arc-length coordinate s at point C,θ_(C)≡θ(s_(C)) is the angle of the arc at point C, and θ_(CD) is theaforementioned angle of the stamp-control system's contact surface; andwherein for each configuration the solution for x_(P) and z_(P) isderived by means of the “shooting method”, whereby an initial value β⁽⁰⁾of β is guessed, the differential equations are solved to yield T(β⁽⁰⁾)and$\lbrack \frac{\partial T}{\partial\beta} \rbrack_{\beta = \beta^{(0)}}^{- 1},$

 Newton-Raphson iteration$\beta^{({n + 1})} = {\beta^{(n)} - {\lbrack \frac{\partial T}{\partial\beta} \rbrack_{\beta = \beta^{(n)}}^{- 1}{T( \beta^{(n)} )}}}$

 is applied to obtain a refined vector β⁽¹⁾, whereupon the differentialequations are solved again; this iteration procedure being appliedrepeatedly until the correct auxiliary boundary conditions T(β)=0 areachieved to within some tolerance.
 12. The apparatus of claim 10 whereinthe trajectory comprises a plurality of configurations, eachconfiguration described by the coordinate s₀ x₀ of the contact front andby corresponding coordinates x_(P), z_(P) of the pivot line given by theequations: x _(P) =x _(E) +R _(s) cos θ_(E) z _(P) =z _(E) +R _(s) sinθ_(E), where x_(E=∫) ₀ ^(L) cos θ(s)ds and z_(E)=∫₀ ^(L) sin θ(s)ds, andwhere the mathematical function θ(s) describing the shape of the arc fora given configuration is assumed to be θ(s)=0 for 0≦s≦s ₀, whereas fors>s₀, θ(s) is determined in stamp segments OC and DE by solution of thedifferential equations ${\frac{u}{s} = {F(u)}},$

the lower-end boundary conditions ${{u_{0} \equiv \begin{Bmatrix}u_{10} \\u_{20}\end{Bmatrix} \equiv \begin{Bmatrix}\theta_{0} \\{\frac{\theta}{s}_{0}}\end{Bmatrix}} = \begin{Bmatrix}0 \\\kappa_{0}\end{Bmatrix}},$

and the upper-end boundary condition${T(\beta)} \equiv {{EI}\frac{\theta}{s}{_{E}{{{{{+ F_{XE}}R_{S}\sin \quad \theta_{E}} - {F_{ZE}R_{S}\cos \quad \theta_{E}}} = 0},{{{wherein}\quad u} \equiv \begin{Bmatrix}u_{1} \\u_{2}\end{Bmatrix} \equiv \begin{Bmatrix}\theta \\\frac{\theta}{s}\end{Bmatrix}},{{F(u)} \equiv \begin{Bmatrix}u_{2} \\{{\frac{F_{x}(s)}{EI}\sin \quad u_{1}} - {\frac{F_{z}(s)}{EI}\cos \quad u_{1}}}\end{Bmatrix}},}}}$

κ₀ is a specified curvature at point O, E is Young's modulus of thestamp, I is the area moment of inertia of the stamp's cross section perunit depth in the y-direction, w is the weight per unit area of thestamp, F_(x)(s) and F_(z)(s) are the x-directed and z-directed stampforces per unit length of stamp in the y direction, given by$\begin{matrix}{{F_{x}(s)} = \{ \begin{matrix}{F_{x0},{0 \leq s \leq s_{C}}} \\{{F_{x0} + {\Delta \quad F_{x}}},{s_{D} \leq s \leq s_{E}},}\end{matrix} } \\{{and}\quad} \\{\quad {{F_{z}(s)} = \{ \begin{matrix}\begin{matrix}{0,{0 \leq s \leq s_{0}}} \\{{w( {s - s_{0}} )},{s_{0} \leq s \leq s_{C}}}\end{matrix} \\{{{w( {s - s_{0}} )} + {\Delta \quad F_{z}}},{s_{D} \leq s \leq s_{E}},}\end{matrix} }}\end{matrix}$

in which F_(x0)≡F_(x)(s₀)≡β is a parameter that is unknown a priori, andthe differences ΔF_(x) and ΔF_(z) are respectively the differences ΔF_(x) ≡F _(x)(s _(D))−F _(x)(s _(C))ΔF _(z) ≡F _(z)(s _(D))−F _(z)(s_(C)) that occur across stamp segment CD where the stamp-control systemcontacts the stamp, the values of which differences, along with thevalue of the difference${\Delta \quad \kappa} \equiv {\frac{\theta}{s}{_{D}{{- \frac{\theta}{s}}{_{C},}}}}$

may be calculated from the three equations of static equilibrium for thestamp under the action of forces applied to the stamp by thestamp-control system, these three differences together with θ_(D)=θ_(C)permitting numerical integration for stamp segment DE to proceedimmediately from the numerical-integration result obtained at the finalpoint C in stamp segment OC; and wherein for each configuration thesolution for x_(P) and z_(P) is derived by means of the “shootingmethod”, whereby an initial value β⁽⁰⁾ of β is guessed, the differentialequations are solved to yield T(β⁽⁰⁾)and$\lbrack \frac{\partial T}{\partial\beta} \rbrack_{\beta = \beta^{(0)}}^{- 1},$

 Newton iteration$\beta^{({n + 1})} = {\beta^{(n)} - {\lbrack \frac{\partial T}{\partial\beta} \rbrack_{\beta = \beta^{(n)}}^{- 1}{T( \beta^{(n)} )}}}$

 is applied to obtain a refined vector β⁽¹⁾, whereupon the differentialequations are solved again; this iteration procedure being appliedrepeatedly until the correct auxiliary boundary conditions T(β)=0 areachieved to within some tolerance.
 13. The apparatus of claim 2 furthercomprising a stamp-control system movable along the x-axis; wherein,throughout the trajectory, each xz position of the upper stamp clamp isa function of the displacement x_(C) of the stamp-control system alongthe x-axis; the trajectory being effective in laying the stamp down uponthe print surface such that the stamp is in continuous contact with acontact surface of the stamp-control system, the location of the contactsurface being characterized by an arc-length coordinate s_(C) thatincreases as the trajectory progresses.
 14. The apparatus of claim 13wherein the stamp-control system is disposed along a line _(C) parallelto the y-axis, line_(C) intersecting the trajectory plane at point Chaving coordinates x_(C) and z_(C), where z_(C) is a fixed, positivez-coordinate during any one printing operation, whereas X_(C) increasesas the trajectory progresses, in coordination with the contact-frontcoordinate x₀.
 15. The apparatus of claim 14 wherein the contact surfaceof the stamp-control system is a plane delimited in the x direction bytwo lines _(C) and _(D) separated by a fixed distance W_(CD), theselines being parallel to the y-axis and intersecting the trajectory planeat points C and D respectively, these points having coordinates(x_(C),z_(C)) and (x_(D),z_(D)) respectively, such that the contactsurface is defined by the three parameters (x_(C),z_(C),θ_(CD)), where$\theta_{CD} \equiv {\tan^{- 1}( \frac{z_{D} - z_{C}}{x_{D} - x_{C}} )}$

is the angle between the contact surface and the print plane, and suchthat the stamp angle θ(s) between arc-length coordinates s=s_(C) ands=s_(D) is substantially equal to θ_(CD); that is, θ(s)≈θ_(CD) for s_(C) ≦s≦s _(D).
 16. The apparatus of claim 15 wherein the upper stampclamp is pivoted about a pivot line _(P) parallel to the y axis andintersecting the xz plane at point P having coordinates x_(P) and z_(P);the stamp attaching to the upper stamp clamp along an upper-clampline_(E) parallel to the y axis and intersecting the xz plane at point Ehaving coordinates x_(E) and z_(E); the upper-clamp line_(E) beingdisposed on the upper stamp clamp at a radius R_(s) from the pivotline_(P), such that the total arc length s_(E) from the lower stampclamp to the line_(E) is s_(E) L, where L is the known, free length ofthe stamp; and wherein the stamp attaches to the upper-clamp line_(E) atan angle θ_(E)≡θ(L).
 17. The apparatus of claim 16 wherein thetrajectory comprises a plurality of configurations, each configurationdescribed by the coordinate s₀ x₀ of the contact front and bycorresponding coordinates x_(P), z_(P) of the pivot line given by theequations: x _(P) =x _(E) +R _(s) cos θ_(E) z _(P) =z _(E) +R _(s) sinθ_(E) where x_(E)=∫₀ ^(L) cos θ(s)ds and z_(E)=∫₀ ^(L) sin θ(s)ds, andwhere the mathematical function θ(s) describing the shape of the arc fora given configuration is assumed to be θ(s)=0 for 0≦s≦s ₀, whereas fors>s₀, θ(s) is determined by solution of the differential equations${\frac{u}{s} = {F(u)}},$

the lower-end boundary conditions ${{u_{0} \equiv \begin{Bmatrix}u_{10} \\u_{20} \\u_{30} \\u_{40}\end{Bmatrix} \equiv \begin{Bmatrix}\theta_{0} \\{\frac{\theta}{s}_{0}} \\F_{X0} \\F_{Z0}\end{Bmatrix}} = \begin{Bmatrix}0 \\0 \\F_{X0} \\F_{Z0}\end{Bmatrix}},$

and the auxiliary boundary conditions T(β)=0, wherein $\begin{matrix}{{u_{0} \equiv \begin{Bmatrix}u_{1} \\u_{2} \\u_{3} \\u_{4}\end{Bmatrix} \equiv \begin{Bmatrix}\theta_{0} \\\frac{\theta}{s} \\{F_{x}(s)} \\{F_{z}(s)}\end{Bmatrix}},} \\{{{F(u)} \equiv \begin{Bmatrix}\begin{matrix}\begin{matrix}u_{2} \\{{\frac{u_{3}}{EI}\sin \quad u_{1}} - {\frac{u_{4}}{EI}\cos \quad u_{1}}}\end{matrix} \\{{{- {p(s)}}\sin \quad u_{1}} - {{f(s)}\cos \quad u_{1}}}\end{matrix} \\{w + {{p(s)}\cos \quad u_{1}} - {{f(s)}\sin \quad u_{1}}}\end{Bmatrix}},} \\{{{T(\beta)} \equiv \begin{Bmatrix}\begin{matrix}{z_{C} - {\int_{0}^{s_{C}}{\sin \quad {\theta (s)}\quad {s}}}} \\{\theta_{C} - \theta_{CD}}\end{matrix} \\{{EI}\frac{\theta}{s}{_{E}{{{+ F_{XE}}R_{S}\sin \quad \theta_{E}} - {F_{ZE}R_{S}\cos \quad \theta_{E}}}}}\end{Bmatrix}},}\end{matrix}$

and wherein F_(X)(s) and F_(Z)(s) are functions of s describing theinternal x-directed and z-directed forces acting on the stamp's crosssection at s per unit depth of the stamp in the y direction,F_(XE)≡F_(X)(s_(E)), F_(ZE)≡F_(Z)(s_(E)), β is a vector of parametersthat are unknown a priori, ${\beta = \begin{Bmatrix}s_{0} \\F_{X0} \\F_{Z0}\end{Bmatrix}},$

s₀ is the aforementioned arc-length coordinate of the contact front,F_(X0)≡F_(X)(0), F_(Z0)≡F_(Z)(0), E is Young's modulus of the stamp, Iis the area moment of inertia of the stamp's cross section per unitdepth in the y-direction, w is the weight per unit area of the stamp,p(s) and f(s) are functions of s describing forces applied normal to thestamp and tangential to the stamp respectively by theprint-force-application system, the stamp-control system and the printsurface, s_(C) is the value of arc-length coordinate s at point C,θ_(C)≡θ(s_(C)) is the angle of the arc at point C, and θ_(CD) is theaforementioned angle of the stamp-control system's contact surface; andwherein for each configuration the solution for x_(P) and z_(P) isderived by means of the “shooting method”, whereby an initial value β⁽⁰⁾of β is guessed, the differential equations are solved to yield T(β⁽⁰⁾)and$\lbrack \frac{\partial T}{\partial\beta} \rbrack_{\beta = \beta^{(0)}}^{- 1},$

 Newton-Raphson iteration$\beta^{({n + 1})} = {\beta^{(n)} - {\lbrack \frac{\partial T}{\partial\beta} \rbrack_{\beta = \beta^{(n)}}^{- 1}{T( \beta^{(n)} )}}}$

 is applied to obtain a refined vector β⁽¹⁾, whereupon the differentialequations are solved again; this iteration procedure being appliedrepeatedly until the correct auxiliary boundary conditions T(β)=0 areachieved to within some tolerance.
 18. The apparatus of claim 16 whereinthe trajectory comprises a plurality of configurations, eachconfiguration described by the coordinate s_(B) x_(B) of the approximatecontact front and by corresponding coordinates x_(P), z_(P) of the pivotline given by the equations x _(P) =x _(E) +R _(s) cos θ_(E) z _(P) =z_(E) +R _(s) sin θ_(E), where x_(E)=∫₀ ^(L) cos θ(s)ds and z_(E)=∫₀ ^(L)sin θ(s)ds, and where the mathematical function θ(s) describing theshape of the arc for a given configuration is assumed to be θ(s)=0 for0≦s≦s _(B), whereas for s>s_(B), θ(s) is determined in stamp segments OCand DE by solution of the differential equations${\frac{u}{s} = {F(u)}},$

the lower-end boundary conditions ${{u_{B} \equiv \begin{Bmatrix}u_{1B} \\u_{2B}\end{Bmatrix} \equiv \begin{Bmatrix}\theta_{B} \\{\frac{\theta}{s}_{B}}\end{Bmatrix}} = \begin{Bmatrix}0 \\\kappa_{B}\end{Bmatrix}},$

and the upper-end boundary condition${T(\beta)} \equiv {{EI}\frac{\theta}{s}{_{E}{{{{{+ F_{XE}}R_{S}\sin \quad \theta_{E}} - {F_{ZE}R_{S}\cos \quad \theta_{E}}} = 0},{{{wherein}\quad u} \equiv \begin{Bmatrix}u_{1} \\u_{2}\end{Bmatrix} \equiv \begin{Bmatrix}\theta \\\frac{\theta}{s}\end{Bmatrix}},{{F(u)} \equiv \begin{Bmatrix}u_{2} \\{{\frac{F_{x}(s)}{EI}\sin \quad u_{1}} - {\frac{F_{z}(s)}{EI}\cos \quad u_{1}}}\end{Bmatrix}},}}}$

κ_(B) is a specified curvature at point B, E is Young's modulus of thestamp, I is the area moment of inertia of the stamp's cross section perunit depth in the y-direction, w is the weight per unit area of thestamp, F_(x)(s) and F_(z)(s) are the x-directed and z-directed stampforces per unit length of stamp in the y direction, given by${F_{x}(s)} = \{ {{\begin{matrix}{{F_{x0},}\quad} & {0 \leq s \leq s_{C}} \\{{F_{x0} + {\Delta \quad F_{x}}},} & {{s_{D} \leq s \leq s_{E}},}\end{matrix}{and}{F_{z}(s)}} = \{ \begin{matrix}{{0,}\quad} & {0 \leq s \leq s_{0}} \\{{{w( {s - s_{0}} )},}\quad} & {s_{0} \leq s \leq s_{C}} \\{{{w( {s - s_{0}} )} + {\Delta \quad F_{z}}},} & {{s_{D} \leq s \leq s_{E}},}\end{matrix} } $

in which F_(x0)≡F_(x)(s₀)≡β is a parameter that is unknown a priori, andthe differences ΔF_(x) and ΔF_(z) are respectively the differences ΔF_(x) ≡F _(x)(s_(D))−F _(x)(s_(C))ΔF _(z) ≡F _(z)(s_(D))−F _(z)(s_(C))that occur across stamp segment CD where the stamp-control systemcontacts the stamp, the values of which differences, along with thevalue of the difference${{\Delta \quad \kappa} \equiv {\frac{\theta}{s}{_{D}{- \frac{\theta}{s}}}_{C}}},$

may be calculated from the three equations of static equilibrium for thestamp under the action of forces applied to the stamp by thestamp-control system, these three differences together with θ_(D)=θ_(C)permitting numerical integration for stamp segment DE to proceedimmediately from the numerical-integration result obtained at the finalpoint C in stamp segment OC; and wherein for each configuration thesolution for x_(P) and z_(P) is derived by means of the “shootingmethod”, whereby an initial value β⁽⁰⁾ of β is guessed, the differentialequations are solved to yield T(β⁽⁰⁾) and$\lbrack \frac{\partial T}{\partial\beta} \rbrack_{\beta = \beta^{(0)}}^{- 1},$

 Newton iteration$\beta^{({n + 1})} = {\beta^{(n)} - {\lbrack \frac{\partial T}{\partial\beta} \rbrack_{\beta = \beta^{(n)}}^{- 1}{T( \beta^{(n)} )}}}$

 is applied to obtain a refined vector β⁽¹⁾, whereupon the differentialequations are solved again; this iteration procedure being appliedrepeatedly until the correct auxiliary boundary conditions T(β)=0 areachieved to within some tolerance.
 19. The apparatus of claim 2 whereinthe print-force-application system comprises a flat-iron.
 20. Theapparatus of claim 5 wherein the stamp-control system comprises a vacuumbar.
 21. A printing apparatus, comprising: a receiver means whosereceiving surface lies in an xy plane, the normal to the surfacedefining a z-axis direction; a lower stamp clamp means for fixing afirst edge of a stamp; an upper stamp clamp means for holding a secondedge of a stamp for movement in the xz directions; a flexible stampmeans for printing to the receiver, said flexible stamp in substantiallythe form of a sheet defining edges, the first edge of which is affixedto the lower stamp clamp, and the opposing second edge of which isaffixed to the upper stamp clamp, thereby allowing the stamp to hang ina curve under gravity and the sheet's own stiffness, such that everynormal to the stamp's curved surface lies substantially parallel to thexz plane; and a trajectory-producing means for moving the upper stampclamp along a prescribed trajectory in the xz plane, such that the stampis draped upon the receiving surface in a manner that causes thecurvature of the stamp near a contact front at a point B to be constantthroughout the trajectory.
 22. The apparatus of claim 21 furthercomprising print-force application means for applying pressure upon thestamp means against the receiver means and for defining the contactfront.
 23. The apparatus of claim 21 further comprising stamp-controlmeans for defining a point C through which the curvature of the sheetwill pass throughout the trajectory.